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THE THEORY OF PULSED FOURIER TRANSFORM MICROWAVE SPECTROSCOPY CARRIED OUT IN A FABRY-PEROT CAVITY

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300 N ZEEBRD ANN ARBOR, Ml 48106
8203415
CAMPBELL, EDWARD JOSEPH
THE THEORY OF PULSED FOURIER TRANSFORM MICROWAVE
SPECTROSCOPY CARRIED OUT IN A FABRY-PEROT CAVITY
University of Illinois at Urbana-Champaign
University
Microfilms
I n t G r n a t i O n a l 300N ZeebRoad,AnnArbor,MI48106
PH.D. 1981
THE THEORY OF PULSED FOURIER TRANSFORM MICROWAVE
SPECTROSCOPY CARRIED OUT IN A FABRY-PEROT CAVITY
BY
EDWARD JOSEPH CAMPBELL
B.A., University of Wisconsin-Madison, 1976
M.S., University of Illinois, 1977
THESIS
Submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Physics
in the Graduate College of the
University of Illinois at Urbana-Champaign, 1981
Urbana, Illinois
•zrrr-i
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
THE GRADUATE COLLEGE
June 2 2 ,
I98I
WE HEREBY RECOMMEND THAT THE THESIS BY
EDWARD JOSEPH CAMPBELL
ENTITLED
THL
TH£0RY
0F
PULSED FOURIER TRANSFORM
MIGROWAVK SPECTROSCOPY CARRILD OUT IN A FABRY-PEROT CANITY
BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
DOCTOR OF PHILOSOPHY
THE DEGREE OF.
UiJ&wt/ 4iMi3iL
Director of Thesis Research
ttt<^Afc(jB
Head of Department
Committee on Final Examination!
Chairman
J 1 "Cfc
%&AfA
lUJZii^
f5 iJjo0^-^
t Required for doctor's degree but not for master's
Ill
The Theory of Pulsed Fourier Transform Microwave
Spectroscopy Carried Out in a Fabry-Perot Cavity
Edward Joseph Campbell, Ph.D.
Department of Physics
University of Illinois at Urbana-Champaign, 1981
A semiclassical theory has been developed to describe
pulsed Fourier transform microwave spectroscopy carried out
in a Fabry-Perot cavity.
A density matrix formalism is used
to study the interaction of a two-level quantum system with
a classical standing wave electric field, appropriate for the
Fabry-Perot cavity.
Equations describing the polarization of,
and subsequent emission of radiation by arbitrary distributions
of molecules in the cavity are derived.
The specific problem
of a static Maxwell-Boltzmann gas is studied in detail, both
theoretically and experimentally.
The static gas lineshape
in the power-broadened limit is described by an ordinary
Doppler and pressure broadened envelope.
Sensitivities of
the ordinary waveguide cell and Fabry-Perot cavity pulsed
Fourier transform spectrometers using static gas samples are
compared.
The gas dynamics of a pulsed supersonic nozzle molecular source are investigated by using the pulsed Fabry-Perot
cavity microwave spectrometer to obtain free induction decay
signals from rotational two-level systems in the gas expansion.
An equation is derived giving the time domain emission signal
IV
lineshape as an integral over the active molecular distribution
in the beam.
detail.
The Doppler splitting phenomenon is discussed in
Experimental lineshapes are deconvoluted to give
molecular velocities, dephasing times, and density distributions.
We find that the density distribution of active mole-
cules from the pulsed nozzle varies rapidly in time, starting
with a depletion on the nozzle axis at short times after the
nozzle is opened, and changing to on-axis concentration at
longer times. Results obtained with the gas nozzle axis
oriented at angles ranging from 0° to 90° with respect to
the direction of propagation of the microwaves are reported.
Rotational assignments are reported for 83KrH35CI,
83
KrD 35 Cl,
i31
XeH 3 5 Cl, and
131
XeD 3 5 Cl.
The measured rare
gas nuclear quadrupole coupling constants, X , are:
XR(MHz)
83
KrH 35 Cl
5.20(10)
83
KrD 35 Cl
7.19(10)
131
XeH 3 5 Cl
-4.64(5)
131
XeD 3 5 Cl
-5.89(20)
The electric field gradient along the a-inertial axis at the
rare gas nuclear site in these four molecules, and in
83
KrH(D)F,
83
KrHC 14 N, and
131
XeH(D)F is found to be directly
proportional to the electric field gradient at that site
calculated from the electric multipole moments of the partner
hydrogen halide.
The proportionality constants are, for
V
83
Kr, 78.5 and for
-24
and -0.12 x 10
go
131
Xe, 160, using values of 0.27 x lo" 24 cm 2
2
cm for t h e nuclear quadrupole moments of
1^1
Kr and
Xe r e s p e c t i v e l y .
This d i r e c t p r o p o r t i o n a l i t y
i s a t t r i b u t e d t o Sternheimer-type quadrupolar s h i e l d i n g of
the r a r e gas nucleus by t h e e l e c t r o n s i n the r a r e gas atom.
Within t h e l i m i t s of u n c e r t a i n t y of t h i s experiment of
£ 0.003 e l e c t r o n we find no evidence f o r charge t r a n s f e r
from
the Kr and Xe atoms.
83
The
Kr n u c l e a r quadrupole coupling c o n s t a n t in
83 ^5
Kr C1F has been measured t o be 13.90 (25)MHz. Using the
known 83Kr nuclear quadrupole moment, t h e f i e l d g r a d i e n t a t
t h e Kr n u c l e a r s i t e i s evaluated and i n t e r p r e t e d i n terms of
t h e quadrupolar s h i e l d i n g constant of Kr, the s t r u c t u r e of
KrCIF, and the e l e c t r i c multipole moments of C1F.
VI
ACKNOWLEDGEMENT
I wish to acknowledge my indebtedness to Prof. Willis
H. Flygare, whose guidance and encouragement made this work
possible.
I would like to thank Dr. Bill Hoke, who taught me
Fourier transform microwave spectroscopy and introduced me
to van der Waals molecules, Dr. Terrill Balle, for the opportunity to use the Fabry-Perot spectrometer, and for discussions
of many of the ideas developed here, and Dr. L. William Buxton,
with whom I cooperated on these, and many other projects.
The
experimental measurements reported in Chapters II and III of
this thesis were done in collaboration with Dr. Balle and Dr.
Buxton.
The microwave spectral measurements reported in
Chapter IV and much of the computer programming work of
Chapters II and III were done in collaboration with Dr. Buxton.
I would like to thank Prof. A. C. Legon for his partici-
pation in the Ar, Kr, and XeHCN work, and for many interesting
discussions of the properties of weakly bound systems.
I would also like to thank Dr. Michael Keenan, who
participated in the
XeD
Cl measurements, Dr. Paul Soper,
Peter Aldrich, Bill Read, Jim Shea, and Dan Wozniak for much
stimulating and enjoyable interaction.
The support of the Department of Chemistry through the
staffs of the electronic shop, machine shop, computer facility,
and student shop, and the Physical Chemistry Secretaries are
gratefully acknowledged.
I would also like to acknowledge
vii
the financial assistance of the University of Illinois.
This
work is based on work supported by the National Science Foundation under Grant 46-32-87-314, title NSF CHE 78-13-7719610,
and by The Petroleum Research Fund, administered by the
American Chemical Society.
viii
TABLE OF CONTENTS
Chapter
Page
I.
General Introduction
II.
The Theory of Pulsed Fourier Transform Microwave
Spectroscopy Carried Out in a FabryPerot Cavity
10
A.
B.
C.
D.
FJ.
F.
G.
1
Derivation of Block-type Equations for
a Standing Wave Electric Field
Appropriate for the Fabry-Perot Cavity . . .
10
Solutions to the Block-type Equations for
the Polarization for Near-Resonant
Radiation Stimulation in the Fabry-Perot
Cavity
17
Evolution of the Molecular Polarization
Following the Removal of the Polarization
Inducing Radiation
28
Electric Field Produced in the FabryPerot Cavity by the Molecules After the
Polarization of the Gas
29
Electric Field in the Fabry-Perot Cavity
Following Near-Resonant Polarization of
Maxwell Boltzmann Static Gas
35
Field Coupling Out of the Fabry-Perot
Cavity
45
S t a t i c Gas Experimental R e s u l t s
49
H.
III.
Comparison of S t a t i c Gas Fabry-Perot
R e s u l t s t o Waveguide Spectrometer R e s u l t s
.
The Gas Dynamics of a Pulsed Supersonic
Molecular Source as Observed with t h e
Fabry-Perot Cavity Microwave Spectrometer
. . .
A.
Introduction
60
65
65
IX
Chapter
B.
IV.
Page
Equations Describing the P o l a r i z a t i o n
and Emission of Radiation by Molecules
in t h e Fabry-Perot Cavity Appropriate
for t h e Pulsed Nozzle Problem
66
C.
Lineshapes in t h e Pulsed-Nozzle FabryPerot Experiment
73
D.
Lineshapes Taken with A l t e r n a t e Nozzle
Geometries
109
Rare Gas Nuclear Quadrupole Coupling in van
der Waals Molecules
, 126
A.
Introduction
126
B.
Experimental
128
C.
Rotational Spectra and Spectroscopic Constants
for the Rare-Gas Hydrogen Halides
130
D. Analysis
146
83
E. Measurement and Analysis of the
Kr Nuclear
Quadrupolar Coupling in 83 Kr 35 ClF
165
185
Appendix A
Appendix B
187
Bibliography
194
Vita
199
Chapter 1.
General Introduction
The study of the pure rotational spectra of small
molecules in the gas phase has since the mid 194O's yielded
a tremendous amount of highly detailed information about both
the structural and electronic properties of these systems.
This information is widely used in the study of molecular
structure and chemical bonding.
It has also been recognized
that the shape, or envelope of a microwave resonance signal
contains much information about the interactions between
individual molecules in the gas phase.
Lineshape studies
have dealt exclusively with a source gas in thermodynamic
equilibrium with its container.
The fundamental aspects of
this problem are now well understood, although with the introduction of sophisticated state-selective techniques and laser
2
technology, the field remains active.
In recent years there have been two significant developments of a general nature that have profoundly affected
both of the problems mentioned above.
The first of these was
the recognition that time domain spectroscopic methods offered
a significant improvement in signal-to-noise over conventional
steady state methods.3 The first time domain experiments in
the microwave region were performed by Dicke and Romer in
1955.4 '5 The pulsed time domain microwave spectrometer as a
standard laboratory instrument was developed by Flygare and
co-workers, beginning in 1975.
Starting from a unified
theory describing coherent transient affects in the microwave
1
and infra-red regions of the spectrum,
these workers investi-
gated a wide variety of coherent transient effects, including
transient absorption and emission, fast passage, Stark switching, and multiple pulse experiments.
A pulsed Fourier Transform
microwave spectrometer incorporating recent developments in
microwave hardware was built and was used to show
that this
technique did indeed offer improvements over the steady state
method.
The spectrometer was used for detailed studies of
13—15
7 14
the relaxation parameters T, and T 2 in NH 3
and OCS,
and it was shown that the time domain method had advantages
in state-selectivity, and was superior to steady state methods
for studying T-, . Advantages in sensitivity and resolution
of time domain work was also illustrated by a study of the
16
molecular Zeeman effect in trans-crotonaldehyde.
A second important spectroscopic development has been
the recognition of the utility of using supersonic expansion
of molecules in an inert, monoatomic carrier gas as a spectroscopic sample.17 These jets provide a cold (1-10 K), nearly
collision free sample, resulting, in the microwave region,
in the elimination of conventional Dopper broadening, of
collisional broadening, and a large increase in signal resulting from the depopulation of the higher rotational levels.
In addition, large numbers of weakly bound van der Waals and
hydrogen bonded molecules can be formed in the jet. Conventional waveguide
methods are unsuitable for studying most
of these systems, except for the more strongly bound hydrogenbonded complexes,18 because of the condensation problem.
3
The first observation of microwave transitions in a van
der Waals complex was that of Novick, Davies, Harris, and
Klemperer,19 who measured several R-branch transitions in
ArHCl in 1973.
They used a Rabi-type molecular beam electric
resonance (MBER) spectrometer, in which a rotational resonance
is detected by observing a change in the number of particles
appearing at the input to a mass spectrometer after the particles
have passed through state selective electrostatic fields.
The gas sample here is a highly collimated, essentially onedimensional, steady state beam that passes at a right angle
to the radiation field. Background pressures ranging from
-5
-9
10
to 10 torr insure that collisional effects are nonexistent.
The first instrument to permit direct observation of
microwave transitions in a supersonic nozzle expansion was the
pulsed Fourier transform Fabry-Perot spectrometer built by
Balle and Flygare in 1979.20 This device incorporates the same
microwave switching and signal handling arrangements of the
earlier Ekkers-Flygare waveguide instrument, but replaces the
waveguide and the static Maxwell-Boltzmann gas sample with a
pulsed supersonic gas expansion in a large volume FabryPerot cavity.
The inherent advantages of time-domain operation,
and the very important advantage of using a supersonic jet
expansion are thus realized in the same instrument.21-23
A block diagram of the Balle-Flygare instrument is
shown in Fig. 1.
We begin by locking the master microwave
4
oscillator (MO) to a harmonic of a frequency standard.
The local
oscillator (LO) is locked 30 MHz from the MO and is used in the
subsequent superheterodyne signal detection.
The Fabry-Perot
cavity consists of two dish-shaped aluminum mirrors, 14 inches
in diameter,
with spherical mirror surfaces.
The mirrors are
mounted so that their separation, and thus the resonance frequency of the cavity, can be adjusted to the molecular resonance frequency.
The mirrors have small coupling holes in
their exact center to allow microwave radiation to be coupled
in or out of the cavity.
The cavity chamber is pumped di-
rectly by a 10 inch oil diffusion pump to maintain a pressure
of 10-5 torr between gas pulses, so that the mean free molecular
path in the cavity is determined by the dimensions of the
vacuum chamber.
The gas nozzle is positioned midway between
the mirrors, just outside their radii, and points directly
at the pump.
We use a General Valve type 8-14-900 pulsed
solenoid valve, with open and close times of 2-10 msec.
valve
The
orifice is simply a flat plate with a 0.020 inch diameter
hole, that is mounted in the bottom of the valve. At some
arbitrary time a 50 volt pulse is used to open the
valve.
The valve is backed by 1-2 atmospheres of a rare gas
seeded with a small percentage (0.5-3%) of another gas, HC1
for example, at room temperature. When the valve opens this
mixture streams into the evacuated chamber, undergoing an
adiabatic cooling process that lowers its rotational and
translational temperatures to between 1 and 10 K.
The gas
molecules travel at approximately 5 • 10 cm/sec and take
Ul
Figure 1. Block diagram of the pulsed Fourier Transform microwave
spectrometer with a pulsed supersonic nozzle gas expansion source.
LO
Frequency
Stabilizer
v - 3 0 MHz
<gfixer
30 MHz
Pin
Diode
£0-
MO
Frequency]
Standard
Frequency
Stabilizer
®
Mixer
Monitor
Detector
D
Pulsed
Nozzle
*^f ,e Pin
Tuner
Circulator
U.
I
Diode
I
f DDetector
e
0HfF^
[Diffusion
Pump
Mi
Mixer
A±30MHz
Mixer
3
>
A
|Computer| - « - • jAveroger|
A/D
Os
Display
Display
7
approximately 1 msec to reach the pump.
At some time during this
transit interval, PIN diode switch 1 is closed for 3-6 msec,
and the gas is irradiated with a IT/2 pulse from the MO.
A
tiny fraction of this incident microwave energy stored in a
nonequilibrium population difference of the two rotational levels
being probed.
Equivalently, the large amplitude driving
electric field in the cavity produces a coherent, macroscopic
polarization of the gas.
The physics of this polarization
process, and of the subsequent emission process, which will be
developed in full detail shortly, is formally equivalent to
the process of pulsed NMR, where now the in- and out-of-phase
components of polarization and the two level population differences replace the more familiar transverse and longitudinal
components of magnetization.
After the gas has been polarized
it will retain this coherent state for a time that is long
compared to the relaxation time of the cavity.
After the power
pulse ringing has died away,PIN diode switch 2, which has
been, protecting the sensitive microwave detector, is closed.
According to Maxwell's equations the polarized gas now sets
up a standing wave electric field in the cavity at the exact
rotational resonance frequency.
This energy is coupled out of
the cavity through switch 2 and is detected in the superheterodyne receiver.
The signal is mixed first with the LO signal
to produce a nominal 30 MHz carrier frequency, and then mixed
down to near-DC with the 30 MHz intermediate frequency signal.
The resulting signal at 10-500 KHz is digitized, averaged,
and Fourier transformed to give the rotational spectrum.
8
The 1 MHz bandwidth of the system is determined by the quality
factor, Q, of the cavity, and is sufficiently large that no
distortion of the microwave signals occurs.
Because we are now observing directly, for the first
time, microwave transactions in a supersonic nozzle expansion,
the microwave lineshapes seen in this instrument are quite
unlike those studied earlier.
As in the static gas,
we need to relate the properties of the experimentally observed
lineshapes back to the properties of the gas expansion from
whence they originate.
This problem is part of the more
general problem of understanding the process of pulsed Fourier
transform microwave spectroscopy carried out in a Fabry-Perot
cavity.
Aside from the unusual gas dynamics, we are dealing
here with standing waves, not the traveling waves of the
waveguide spectrometer, and the theory must also take into
account the high-Q, narrow banded characteristics of the
Fabry-Perot cavity.
In Chapter II of this thesis I will
develop the equations appropriate for a semiclassical description of time domain spectroscopy in the Fabry-Perot cavity.'
This initial development will be quite general, and not fixed
to any particular distribution of gas molecules in the cavity.
Because this theory does yield specific predictions for the
use of a Maxwell-Boltzman distribution of molecules in the
cavity, this static gas problem will be investigated, both
theoretically and experimentally.
This will allow me to draw
out some of the features of the high-Q, standing wave solutions
in this somewhat more familiar situation, and also to develop
9
a contrast between the cavity results and the waveguide experiment.
In Chapter III I will apply the cavity equations to the
problem of a pulsed supersonic gas expansion.
This work
will establish the basic relationships between the observed
25
lineshapes and the properties of the gas expansion.
The
intent here is first, to understand the properties of the gas
expansion, and second, to provide a systematic characterization
of the spectrometer operation in order to allow us to make
optimum use of it in studying chemical problems.
In Chapter IV I will present measurements of the rare
gas quadruple coupling constants in four rare gas-hydrogen
halide systems.
These results are combined with similar
measurements for five similar systems to obtain a systematic
explanation for the magnitude of these coupling constants in
terms of the geometry of the van der Waals molecules, the
electric multipole moments of the hydrogen halides, the quadrupolar shielding properties of the rare gas subunits, and the
nature of the van der Waals bond.
10
Chapter I I .
A.
The Theory of Pulsed Fourier Transform Microwave
Spectroscopy Carried Out in a Fabry-Perot Cavity.
Derivation of Bloch-type equations for a standing wave
electric field appropriate for the Fabry-Perot cavity
Consider an ensemble of non-degenerate two-level quantum
systems interacting with an electric field E(r,t) through the
electric dipole interaction.
In the absence of collisions,
the quantum-mechanical Hamiltonian for the j ' t h molecule i s
H(r.,t) = H - (-n2/2m)V2 -
u
• E(r.,t)
(1)
where r . i s t h e c e n t e r - o f - m a s s o f t h e j ' t h m o l e c u l e , H„ i s t h e
t i m e - i n d e p e n d e n t f r e e m o l e c u l a r H a m i l t o n i a n whose e i g e n f u n c t i o n s i n c l u d e e n e r g y l e v e l s E a n d E, c o r r e s p o n d i n g t o t h e
upper a n d lower s t a t e s r e s p e c t i v e l y , V i s t h e g r a d i e n t w i t h
r e s p e c t t o r . , m i s t h e m o l e c u l a r mass, a n d y i s t h e d i p o l e
moment o p e r a t o r .
Since t h e e x t e r n a l f i e l d i n t e r a c t i o n does not
a f f e c t t h e molecular center-of-mass motion, t h e density matrix
a"1 ( r . , p . , t ) i n c l a s s i c a l phase s p a c e , where r . a n d p . a r e t h e
~j ~J
~J
~j
c l a s s i c a l v a r i a b l e s f o r t h e p o s i t i o n a n d momentum of t h e j ' t h
m o l e c u l e , s a t i s f i e s 26
^ ^ " V ^ a a ' ^ j ' P j ' ^
"
t V r ^ j ' ^ ^ j ' P j ' ^ a a '
»)
where t h e b r a c k e t s on t h e r i g h t h a n d - s i d e i n d i c a t e t h e commutator.
Provided t h a t each molecule i n t e r a c t s
with t h e f i e l d ,
independently
we may work w i t h a m a c r o s c o p i c d e n s i t y m a t r i x
26
having elements defined by
,(r,p,t) = Z/d3r.d3p aD
a
.(r.,p.,t)6(r-r.)5(p-p.). (
Using v = p/m, Eq. (2) becomes
m t + v y ) a m l ( r , v , t ) = [H -y-E (r,t) ,0 (r,v,t) ]
,,
which we take as the starting point for this work.
Before taking up the standing wave solutions to this
equation it will be useful to review the traveling wave reg
suits. The development here will differ only slightly from
that in the McGurk, Schmalz, and Flygare paper.
We take
E(r,t) = 2z ecos(cot-ky) , a plane polarized traveling wave,
where z is a unit vector perpendicular to the direction of
propagation, and where £ may in general have the form e(r,t).
Using this, Eq. (4) becomes
iii(
!t
+
r ! ) a a a > " ^ ^ b a ^ b ^ a b ^ b a 5 coa( U t-ky)
±R(jL. + Y - ^ a b ^ a b ' W
(5)
+ 2lJ
itt(
lt
+
Y*! )a bb
= 2£
ab e ^ a a - c W
cos(ut-ky)
^ab a ba"^ba°ab ) coB< M t-ky).
Following McGurk, et al., we transform to the density matrix
in the interaction representation defined by
p = exp(^(t-J)aexp(-^- (t-£)),
(6)
(
12
where
E.
(7)
S »
0
E&
+hu
Substituting p for a in Eq. (5) and making the rotating wave
approximation, 6 one obtains
9t
+
Y-!>Paa
=
e
i,h{
3t
+
Y'! } Pab
=
-*AwPab ^ a b ^ a a - P b b *
+
Y'^Pbb = ^Wba-lWW
(10)
- w(l-v/c),
(11)
^It
^baPab^abPba5
(8)
i<ft(
(9)
where
Aw = u
and
VBa
w
o'T -
'
the two-level resonance frequency.
relationship k = u/c.
in McGurk et al.
dependence.
vV
We have assumed a dispersion
To make the connection with Eqns. (17-19)
we specify e(r,t) = e (t), with no spatial
With no r dependence in Eqns. (8-10) above, the
terms may be handled trivially by specifying that p
= p. , (v,t) .
0606
**
One is left with
, (r,v,t)
13
3p a a
^ T T " ^baPab'Wba*
3p
411
x
ab
TF"
-
* T t "=
^ab^aa'Pbb^^Pab
(12)
e (y
abPba"libaPab> *
These may be compared to Eqns. (17-19) and (65) in McGurk et al.
Consider now the standing wave case. The electric fields
27
in the cavity
in the TEM
modes used in this experiment
will be polarized Gaussian standing waves of the form (See
Fig. 2)
E(r,t) = 2ze(r,t) cos wt .
(13)
We transform to the density matrix in the interaction representation defined by
p = exp(^p t)a e x p ( ~ t ) ,
where S i s defined in Eq. (7).
(14)
Again making the rotating wave
approximation one obtains
ifl(
lt
+
Y'^Paa
=
- e ( £ ' t } ^abPba^baPab 5
(ft
+
Y'^Pab
=
^'^ab'Paa-Pbb*
<ft
+
Y-^Pbb
=
- e ^ ' f c ) ^baPab^abPba)
ift
ift
" *Aupab
<15>
where
Aw = w0-w,
(16)
Figure 2. Geometry and coordinate system used for the Fabry-Perot
cavity. The distance at which the electric field falls
to 1/e of its on-axis value is w(y), with w(y = 0) = w ,
the minimum value. The radius of curvature of both
mirrors is R = 83.8 cm.
15
16
with no v/c term, since the transformation a+p involves no
spatial dependence.
The effect of molecular motion on the
standing wave solutions enters via the v'V terms.
This point
will be discussed later.
Following closely the development in McGurk et al.
for the traveling wave we rewrite Eq. (15) in terms of the
polarization and population difference of the gas.
The polari-
zation is given by
P = Tr(ya),
(17)
where Tr represents the trace of the matrix representation.
Using Eq. (14) gives
P = Tr(y exp(-|£ t)p exp(^- t)) .
(18)
Expanding this, one obtains
* - ^baPab ^
+
»abPba ^
= (Pr+iP.) e la)t + (Pr-iPi) e" i w t ,
<19>
(20)
with
(
ViPi)
=
^baPab
by definition, and where P r and P. are real quantities.
ing that N a = p
<21)
Not-
, N b = P b h , the number densities of molecules
in the a and b states respectively, from Eqns. (15), (19),
and (20) one obtains the Bloch-type equations for the standing
wave case:
17
P
3
(
ft
+
(
ft
+
)P +AwP
Y'! r
i
+
r
T^ = °'
(22)
pt
a
)p
2
Y*! i " ^wP r +K e(r,t) (^M) + ^ i = 0,
*AM
Here AN = Na-Nj3, AN
(23)
* (AN-AN )
is the equilibrium value of AN in the
absence of an electric field, and K = 2fi
|<a|u |b>|. The
first-order relaxation terms involving T, and T 2 have been introg
duced phenomenologically in the usual way.
B.
Solutions to the Bloch-type equations for the polarization
for near-resonant radiation stimulation in the Fabry-Perot
cavity
Our problem now is to solve Eqns. (22-24) for P.,Pr,
and
N using functional forms of e(r,t) appropriate for the
pulsed Fourier transform experiment carried out in a FabryPerot cavity.
work,
'
The reader is referred to previously published
detailing the pulsed Fourier transform technique.
We are interested in applying these equations to two
different kinds of problems.
nozzle experiment.
Consider first the pulsed
In this case it is possible to define for
each coordinate r a unique velocity v(r), since it is known
that all molecules travel at constant speed along straight
line paths originating at the nozzle opening.
Eq. (22) by dv, and integrating, one has
Multiplying
18
IT- /dv Pr(r,v,t) + / dv v V P(r,v,t) +
Aw ; dv P.(r,v,t) + -L / dv P„(r,v,t) = 0.
»V
J.
rv
INS
A »\
^*
i
*M
*«
This is
9P (r,t)
——n
+ V • / dv v P r (r,v,t) +
Aw P, (r,t) + ^- P r (r,t) = 0.
l ~
r ,,
i2
Because each coordinate
(25)
has a unique velocity associated
with it, the second term is
V
• / dv v P„(r,v,t) = V • (v(r) P^ (r,t)) .
Now define
Pr(r,t) = N(r,t) pr(r,t)
P. (r,t) = N(r,t) p. (r,t)
(26)
AN(r,t) = N(r,t) An(r,t)
where
N (r,t) = Z/dv p
a
-
aa
(r,v,t) ,
~ -v
the number density of two-level systems at coordinate r at
time t.
Eq. (25) becomes
9p
r
3N
N -rr + P„ |z- + P- V • (v(r)N) + v(r)N • Vp„
Np r
+ AwNp. + -==• - 0.
(27)
We will assume a conservation law for N of the standard form
19
9N(
f l t ) + V • (v(r)N (r,t)) = 0.
Dividing the remaining terms in Eq. (27) by N(r,t), and repeating this entire procedure for Eqns. (23) and (24), one obtains
(
(
p
9
3t
+
( }
p
Y 5 'V
3t
+
Y ( £ } *!) p i "
a
r
+ Awp
Awp
r
i
+
T^ = ° '
r
+ <2 e(
£' t) ^x L + r 1
,((««
= 0/
(28)
-n(An-An )
< J L + y ( r > . v ) * A n - E ( r , t ) p . + — T ^ - 2 - - 0,
where An for a given molecular transition will depend only on
the Boltmann factors, and will be assumed to be a constant.
Now consider the static gas problem.
a constant.
Here N(r,t) = N,
We may cast Eqns. (22-24) into a form similar
to Eq. (28) by defining
P„(r,v,t) = NW(v) p„(r,v,t) ,
•*•
«S*
*V
Al
X.
(29)
*s, <\«
with analogous expressions for p. and An. The normalized
Boltzmann distribution of velocities is given by
\
w
W{
1v
= l(_JL_)V2
Y'
2irkT;
e
-mv2/2kT
~
Using Eq. (29), Eqns. (22-24) are, for the static gas problem
< £ + v -V) pr + Awp. + |j= 0 ,
(
jt
*
+
Y 'V
p
i "
,»«„
Aa)p
r
+ <2 e
^'t)1T1
+
*r = ° '
«fi(An-An )
(30)
20
This is identical to Eq. (28), except v(r) is replaced by the
phase-space vector v, and pr(r,v (r),t) is replaced by
P r (r,v,t), with similar exchanges for p. and An.
For the so-
lutions to Eqns. (28) and (30) to be developed here it is
useful to note that for any velocity field satisfying the
criterion t
v(r) • V v. = 0,i = x, y, z
(31)
solutions to Eqns. (28) and (30) may be exchanged by making a
formal substitution p„(r,v,t) ->- pv.(r,v(r) ,t) , etc.
Since the
pulsed nozzle molecular velocity field satisfies Eq. (31),
the terms v and v(r) in the solutions p , p., and An to be
«- ~
•v
r
i
developed below may be freely interchanged according to whether
one is dealing with the static gas or with the pulsed nozzle.
We first note the existence of formal solutions to
Eqns. (28) and (30). One may combine the first two equations
in (30), for example, as
(
3t
+
VV
^r"*"1?^ " 1 Aw(p r +ip i )
(32)
2
ttAn
<?_+!?;,)
r
1
+ iK 2 e<r,t)^p +
= 0.
The formal solution to this is 28
(pr+ip.) (r,v,t) = / d t - ( - i 4 J W i A t t " l A 2 M t - t , > x
1
~ ~
-co
4
e(r-v(t-t') ,t')An(r-v(t-t') ,v,t') ,
See Appendix A
(33)
21
with a similar result
(An(r,v,t) - AnJ = / dt' (i)e" (t_t ' ) / T l
x
(34)
e(r-y(t-t'),t') pi(r-v(t-t'y, ,v,t') ,
for the population difference. Using Eqns. (33) and (34), the
solutions for the polarization and population differences may
be developed iteratively to any desired degree of accuracy, a
process entailing considerable mathematical complexity. 29 It
will be possible to adopt here a simpler approach still capable
of explaining the phenomena occurring in the Fabry-Perot
cavity experiment.
Consider an experiment in which the molecules in the
cavity are brought into contact with a radiation field
2e(r,t)cos wt for a duration T , where e(r,t) satisfies the
cavity boundry conditions.
We consider two sets of solutions
to Eq. (30). In the case where KET
<ff/2throughout the
central part of the cavity, and where kvi << 1, where v is a
characteristic molecular velocity component along the cavity
axis, a typical molecule during the pulse time T travels only
a short distance compared to the characteristic distance k"
of changes in the intensity of the standing wave electric
field.
obtain
One may then ignore the v • V terms in Eq. (30) to
22
3P r
Pr
-gt+AWPi + ^ ^ O ,
P
i
. „ . 2 . . . «nAn . P i _ n
- g £ " • AooPr + K e ( r , t ) - j - + ^ - = 0,
3tT"
e(r,t)Pi + — ^
,, CN
(35)
=0.
These equations are similar in form to Eq. (46) in McGurk et
al.
, except t h a t now p_, p . , and
—
X
n are functions of r as
X
*v
well as t, and Aw here contains no velocity dependence.
When
the two-level system is brought instantaneously into contact
with a radiation field of constant amplitude, so that e(r,t)
may be written e(r), one has
p
i(^
2
K
*ne(r)An
t) =
—w:
, ,.
-K2 ,fled:)An
.
P r (-,t) = —^
£ f
r
4i
~
2
"
An(r,t) = An
°
_t/T
e
2
(cosftt - ftT^sinftt)-1
7TZT72
2
d/T2r + n
e" ^ 2 (OT0cosftt + sinftt)-OT0
\
±
2
(1/T 2 )' + or
(ice (r)) 2 e"" t/ ^2 (cosJ2t+(OT2) _ 1 sin^t) + (1/T2) 2+Aw2
*
5
(1A2> + «
where
2
ft =
(assuming T,=T_)
2 2 , v
2
Ke
( r ) + Aw
(36)
23
These expressions can be simplified under certain experimental
conditions.
The electric field e(£) inside the cavity appearing in
these expressions can be related to the input power to the
cavity by making use of bulk circuit concepts.
The quality
factor Q is defined for the Fabry-Perot cavity by
Q _ 2TT (total energy stored)
(energy dissipated per cycle) .
Four Q's are defined for the cavity:
,3?*
'
Q , Q - / Q c 5' a c c o u n t i n g
for power dissipation in the cavity alone, in the input coupling,
and in the output coupling, respectively, and Q_, which combines all the power dissipating elements, where
1/QL = 1/QQ + 1/Q cl + V Q c 2 .
We will assume that Q
= Q j = Q„2'
(38)
I n t h e cav
one n a s
^Y
a
maximum electric field strength given by 21
1/2
-2
i
f
(39)
,u Q cl d w o
where
R = total available power at the input coupling,
d = mirror spacing
w
= beam waist parameter.
Taking R = lmW, Q L = 1-104, w = 2Tr'12«109s_1, Q
±
= 3«10 4 ,
d = 47 cm, and w = 6 cm, values typical for our experiment,
_3
one o b t a i n s E = 6-10
e s u , o r , p i c k i n g up t h e f a c t o r o f
2 from Eq. ( 1 3 ) , e Q = 3 ' 1 0 ~ 3 e s u , and Ke o /2ir = 1*10 Hz f o r
24
a typical 1 debye electric dipole moment.
The bandwidth that
one can work with in the Fabry-Perot experiment is limited by
4
the cavity bandwidth, Av c . For a loaded Q of 10 at 12 GHz
one h a s
.
v _ 12*10 9
Av_ = pr- =
2
C
U
L
=
, 0.,n6 „
1«2'10
Hz.
10*
This f i x e s a maximum Av of 0.6*10
Hz.
Our experimental work
i s a c t u a l l y generally l i m i t e d t o frequency o f f s e t s Av < 0.3*10
Hz, and t h i s w i l l always be the c a s e in experiments discussed
in t h i s paper.
Over most of the c e n t r a l region of t h e cavity
then, excluding the nodal surfaces of the standing wave
f i e l d , one may invoke t h e condition (Aw = 2irAv)
A
2
^—5- «
1.
(40)
I t w i l l be shown l a t e r t h a t most of the molecular emission i s
produced in t h e h i g h - f i e l d , a n t i n o d a l , c e n t r a l region of the
c a v i t y , so t h i s approximation i s a good one.
2
2
2 >> T —2 , and
In the c a s e where ( K C
o J >> Aw , U Oo
^9
where t h e p o l a r i z i n g p u l s e width T s a t i s f i e s
P
T
K<
K< T
(41)
P
7"f ' T P
2 '
r
^
Aw
conditions easily met in the pulsed nozzle experiment, Eq.
(36) reduces to
25
P
i(5't)
Pr(r,t)
=
=
K-nAn
4~^
sin(< e
(r)t)
K-RAn .
j-°- £ | [cos(Ke(r)t)-l]
(42)
A n ( r , t ) = AnQ c o s ( < e ( r ) t ) .
If -^— << 1, then p may be neglected in comparison to p..
Ke
o
The resulting expressions for p , p., and An satisfy
X
3Pi
+ K
3t
3
2 e& (v r ) hAn
* t'
4
/ »nAn ^
=
0
,.
e
X
(43)
n
p
at I T - ; - <£) i - °'
with the initial conditions
p <r,t=0) = p.(r,t=0) = 0, An(r,t=0) = An .
X
*v
X
»«
-«
(44)
o
Following the same considerations that led to Eq. (43),
a second set of solutions to the polarization Eqns. (28) and
(30) are obtained by setting Aw equal to zero, and dropping
the terms involving T, and T 2 .
The resulting equations
(tr: + v • V)p r = 0 ,
(ft + v * V)p. + K2e(r,t) ^f-
(
(45)
= 0,
ft + Y ' V T 2, - e^'t^i = °'
(46)
<47>
26
2
will be valid for (KeQ)
2
—2
(KG 0 )
>> T 2
, and T
2
>> Aw , T
<< T 2 .
2
(<e )/Aw , and when
«
Alternately, Eqns. (45-47) may
be regarded as a microscopic description of on-resonant
polarization.
The exact solutions to Eqns. (45-47) with the
initial conditions in Eq. (44) are
Pr(r,y,t) = 0
(48)
iChAn
t
— T - 2 sin<K:/e(r-v(r)(t-t'),t')dt'),
P,(r,v,t)=X
«v
<v
"
,
t
t
An(r,v,t) = An cos (K /
t
r%,
«%»
O
*v
*v
(49)
*v
O
e (r-v(r) ( t - t ' ) , t • )dt') .
*V
*V
(50)
*W
Here to is defined as the time at which the electric field
e(r,t) is switched on.
We require that the velocity field
v(r) satisfy v • Vv- = 0,
where i = x,y,z.
The compatibility
of solutions (48-50) to the general result in Eqns. (33) and
(34) may be verified by direct substitution.
Comparison of
Eqns. (48-50) to Eq. (42) shows that the terms e(r)t in
Eq. (42) have been replaced by
/ e(r-v(r)(t-t«),t')dt'.
%
" -
(51)
-
When e(r,t) = e(r), and a molecule travels only a short distance compared to the distance scale of changes in e(r),
Eq. (51) does reduce to the simpler expression.
r
See Appendix A
In general,
27
subject to the previously listed conditions, the quantities
p. (r,v,t) and An(r,y,t) depend only on the total integrated
electric field envelope experienced by the molecules as they
move through the spatially complex, time varying fields of the
.. 30
cavity.
We will not work here with that part of the polarization
and population expressions (49) and (50) resulting specifically
from the v • V terms in Eqns. (45-47).
Experimentally, we find
no unusual behavior in the emission line shapes obtained from
long polarization times (i.e., under conditions where v • V
may not be set to zero in Eqns. (45-47)).
Because it is al-
ways possible to work with optimum signal-to-noise in the short
pulse limit, and because the additional algebraic complications
of any long pulse results, which must be handled numerically,
will not provide a significantly more helpful description of
the operation of the spectrometer, we omit any attempt to
provide a description of the long pulse limit.
Accordingly,
Eqns. (49) and (50) are simplified to
KftAn
t
j - 2 s i n (</e(r,t')dt') ,
c
o
p. (r,t) =
An(r,t) =
An Q
t
cos UJe(r,t')dt'),
fc
o
(52)
(53)
strictly valid in the short pulse limit defined by
kvx
<< 1 ,
f o r t h e c a s e when
KE„T„ <
o p ~
i p <<(kvKe0)~1/2
(54a)
1 , o r by
•*
(54b)
28
when ice T
> 1.
Relationship (54a) has already been discussed
in connection with Eq. (31). Equation (54b) follows from the
fact that we are working here with standing waves, and will be
derived in Part III.
We mention finally, that the on-resonant solutions to
Eqns. (28) or (30) in the presence of a simple first-order
relaxation mechanism with T. = T» may be recovered from Eqns.
(48-50) by regarding t
as a time of last collision and per-
forming a weighted average over all possible such collision
times.
C.
Evolution of the molecular polarization following the
removal of the polarization inducing radiation
After the system has been polarized, the input radiation
is switched off and the molecules emit.
Let the polarizing
radiation at frequency w be removed at time t, .
At that
instant one has a polarization (from Eq. (20)) given by
P(r,v,t,) = (P(r,v,t,) + iP. (r,v,tn) )eltotl + c.c. ,
«w
«u
X
X * * * s * X
X
*
v
*
w
(55)
X
where 'c.c' stands for complex conjugate.
The values P (r,v,t,)
and P.(r,v,t,) are determined by solving Eqns. (28) or (30)
1
*v <v
X
for t < t , and then using either Eq. (26) or Eq. (29). At
times greater than t., the evolution of the quantitites p.,
p , and An is described by Eqns. (28) or (30) with e(r,t) in
those expressions set to zero.
(
These are
ft + Y"! )p r + A w p i + T" = °'
(56)
29
p
,a
(ft
+
•s
)p
Y'! i '
AwP
J«.A„
r
+
i
T^ = °'
(57)
'MAn-AnJ
<h+ rz> Y + —wf 2 - - °-
<58>
The solutions to these equations with arbitrary initial conditions at time t, are:
p (r, v, t) = e" ( t - t l ) /*!{*
(r-v (r) (t-t,), tn) cosAw (t-t,)
(59)
Pite-vlx)(t-tx),t1)sinAw(t-t1)>,
-
P4(r,v,t) = e " (t-t l )/lr 2 {p. (r-v(r)(t-t1),t1}ooBflw(i?-t1)
X
*v <**»
X
«* *v *w
X X
X
(60)
+ pr(r-v(r) (tr-t1),t1)sinAw(t--t;L)},
An (r,v,t) = An + e"( t ~ t i^ T i {An (r-v(r) (t-t.), t,) -An }.
(61)
These expressions may be compared to Eq. (132) in McGurk
et a l .
Again w e require that the velocity field satisfy
Eq. (31).
D.
Electric field produced in the Fabry-Perot cavity by
the molecules after the polarization of the gas
The wave equation for the electric field in the cavity
produced by the polarized molecules is, from Maxwell's equations ,
(V2 - iy-4 " H ydZ
or at*
cT
8 = *Z^4-4
"
c* St*
4uV(V'P),
~ ~~
(62)
30
where a is the conductivity of the medium in the cavity, and
P is given by expressions (20), (59), and (60), along with
Eqns. (26) or (29) , depending on which problem is being considered.
Consider first the V(V#P) term.
radiation of the TEM
For the z-polarized
cavity modes (Eq. (13) and Fig. 1 ) ,
where P = P z, this is
!«!•!> " l i f e ; + wfs" ^ + &-; •
(63)
The electric field arising from the x and y terms in Eq. (63)
is not coupled out of the cavity, so those terms are ignored.
2
2
-2 2
2
The final 9 P/3z
term will be small compared to c
3 P/3t
as
long as P varies slowly along the transverse, z-axis in the
central region of the cavity.
For the TEM
modes to be used
here this requires that the applied polarizing electric field
not be so large that molecules in the center of the cavity are
driven past the condition for optimum emission of radiation.
Because this requirement is met when the cavity is operated at
optimum signal-to-noise, and since we observe no unusual effects
in lineshapes obtained under conditions of either high polarizing electric fields, or operation in modes other than the
TEM
, we will ignore this last term as well.
The remaining
equation must be solved with the boundary conditions appropriate
for the Fabry-Perot cavity, and in this experiment, with a set
to zero. Rather than follow this last procedure however, we
29
will adopt the method of Lamb
and adjust the a term in Eq.
(62) to account for the cavity damping.
The cavity time
31
c o n s t a n t Tc i s defined by
(64)
Tc = QL/w,
where Q
includes coupling, reflection, and diffraction losses
in the cavity.
Then
<v* - h —2
c* 3tz
h — > E = H- ~~2 •
xcc
3t
~
< 65 >
c* 3t z
Rewriting Eq. (20) using Eqns. (59), (60), (26), and (16), one
has (choosing the pulsed nozzle case)
P = N(r,t)
{(pr • + ipl ")e i ( w o t + ( u " a , o ) t l > + c.c.}
*\#
(66)
u / / )/T
w i t h p ' = e —v (t—t
l i 2 p _ ( r - v ( r ) ( t - t , ) , t . ) and an analogous
X
JL ~
expression for P.;'•
~
~
X
X
,
We will assume a solution to Eq. (65) of
the form
E(r,t) = (e +ie,)u(r)e i ( w o t + ( w - w o ) t l ) + c.c.
-v
X
X
(67)
~
where e , e . , and u(r) a r e r e a l , and e
and e. a r e slowly vary-
ing functions of time,
(V2 + (-£> 2 u(r) = 0,
(68)
where w„ i s t h e resonant frequency of the c a v i t y ,
' c a v i t y *(Vu<VdV
- l>
and u(r) s a t i s f i e s the c a v i t y boundary c o n d i t i o n s .
i n g Eq. (67) i n t o Eq. (65) and ignoring a l l terms
<69>
Substitut-
32
9e r 3 2 e r 3e i 3 2 e i 3p' r 3 2 p' r 3pi« 3 ^ ' 3N 32N
3t, 3t 2 , 3t ,3t 2 , 3t , 3t 2 , 3t, 3t 2 , 3t, 3t 2 ,
one obtains
(er + i e ^ V ^ r ) + (-§)2u(r) (e r +i £i )
iw_
^2
4TTW
(er+ i e i> u ( r >
T ~ (p r' + iPi')'
=
~
TCC
< 7 °)
c
plus the corresponding complex conjugate equation.
Equating
the real and imaginary part of this expression, multiplying
through by u(r), and integrating over the cavity, one has,
using Eqns. (64), (68), and (69),
w
c 2 'u(r)N(r,t)dV
/p^utoNOr^dV - 0^ (l-(^n/p
r
e
= 47TQ.
r
-
L
2
2„ ,V2,2
OL^ST* >
o
l—I
,
(7i)
.
(72)
+ 1
w
Jpr 'u(r)N(r,t)dV
+ Q.ii ( l - w^c n2/ p ,i 'u(r)N(r,t)dV
~
~
~
~
e
= -4-nQ,
1
2
•"
?
w
c 2 2
Q 2 a-(-fn 2 + i
L
wQ
In the1'Fabry-Perot cavity the frequency width of any
single mode is always much less than the distance to the nearest nondegenerate mode. Our work here will therefore be confined to single-mode operation. The solution for u(r) in the
27
TEM^
ooq mode is
U(
^
= u
w
2 2 2
2 2
e ( X + Z > / W {y)
x
o w7yT "
cos(ky+k( ^-)-<(,-Trq/2), (73)
33
where we use the coordinate system of Fig. 2, and
w
c
k = —
c
,
'
wrt = (=^(d(2R - d ) ) 1 / 2 ) 1 / 2 ,
O
O
2lT
'
(74)
2 1 2
w(y) = w 0 ( l + ( - A Y _ ) ) /
,
7TW 0
<>j = t a n " 1 (-^-%) ,
7TW0
A is the free space radiation wavelength, d is the mirror separation, R
is the radius of curvature of either mirror, q + 1
the number of half-wavelengths between the mirrors, and u
is
determined by the normalization condition in Eq. (69). The
beam waist, w , is the distance from the center of the cavity
to the 1/e points of the field strength.
The Gaussian beam
diameter expands from a minimum value of 2w
at y = 0 to
2w(y = d/2) at the mirror, typically an increase of 20%. For
our purposes here the form
2
2
u(r) = u 0 e" p / w o cos (ky- -rrq/2)
2
will suffice.
Here
p
2
= x
(75)
2
+ z , and w(y) has been set to
w . Substituting Eq. (75) into Eqns. (71) and (72) and using
Eqns. (59), (60), and (48) one obtains for the pulsed nozzle
in this case:
34
2 2
e"P/wo u 2
i—
2_
2
2 2
a (i-(-£) ) + i
"L
wQ
STTO.
E(r,t) =
cos(ky-ir(q/2)) e
-(t-tO/To
x
** x
/ v d 3 r ' p ( r ' - v t r ' J t t - t , ) , ^ ) exp(-p' 2 /V 2 ) cos(ky'^Tq/2)N(r',t) x
{ooBfc^fcffcaTo^t^ + Q j U - C - ^ n sin(w 0 t+(w-to Q )t 1 )}
(76)
.
For the static gas problem, replace v(r') by v, and perform an
additional integration over dv, weighted by the normalized
Boltzmann distribution, W(v).
Boundary conditions for the cavity are incorporated in
this result in the normal mode form u(r) outside the r' integral, and in the Q_, cos(ky'—rrq/2), and exp(-p'2/w 2) terms
inside the integral. A radiating dipole placed between two
mirrors can build up an electric field enhanced by a factor
of Q_ only if it is positioned so that successive reflections
of the emitted field do not destructively interfere. As the
dipole moves off axis the effectiveness of the mirrors in
2
2
H
/w
intercepting the emitted field is reduced by the e-o'
' o
term. Notice also that the presence of the normal mode over
the integral in Eq. (76) weights the polarization in favor of
that produced in the central, antinodal, and thus high-field
portions of the cavity.
Comparing Eq. (76) to Eq. (138) in
McGurk et al. we find that the argument
w ^ t - t ^ + w(v/c) (t-t^-kz + wtj^
(77)
35
in Eq. (138) has become separated i n t o w ( t - t . ) + wt, o u t s i d e
t h e i n t e g r a l i n Eq. (76), and
[k] (r'-v(r')(t-t,))
* * * * * *
(78)
X
inside the polarization term.
This separation is analogous to,
and results from the separation of cos(wt-kz) into cos wt cos
kz that occurs in going from a traveling wave to a standing
wave electric field.
Expression (77) may be understood as a
Doppler shifted radiation frequency of m
+ in (v/c) , and expres-
sion (78) as a phase shift resulting from the movement of a
dipole from a region excited with one phase to another region
5
where the dipoles have a different phase.
E.
Electric field in the Fabry-Perot cavity following nearresonant polarization of a Maxwell Boltzman Static Gas
We are now ready to discuss pulsed time-domain spectro-
scopy carried out in a Fabry-Perot cavity.
The work in this
section will be limited to the static gas problem.
This will
allow us to draw out some features of the high-Q, standing
wave solutions from the previous sections common to both the
static-gas and pulsed nozzle experiments while still treating
a familiar problem.
The pulsed nozzle and the resultant gas
dynamics will be taken up in the following chapter.
Consider now the static gas experiment.
microwave pulse of typical duration T
A high power
«, 1 us is applied to a
tr
gas sample in the cavity at thermodynamic equilibrium and at
a pressure of several millitorr or less.
Our discussion is
36
of course limited to the region Aw <<<e
as discussed earlier,
although this restriction does not appear to be important
experimentally.
To help simplify some later results, we will
for convenience also specify w = w
and Aw <<Aw_, that is,
c
c
all polarization and emission processes are to be carried out
well within the cavity bandwidth, with the cavity tuned to
the carrier.
Polarization of the gas is described by Eqns.
(48) and (52) . Using the assumption that w = w„, we take
e(r,t) (Eq. (13)) to be
-(x2+z2)/w
e(r,t) = e e
°
2
-t/x
(1 - e
) sin ky
(79)
for 0 <_ t <_ T ,
P
and
£(r,t) = £ y
(x2+22)/w
o2(l-e*Tp/Tc)exp(„(t.Tp)/T^sin ky
f
for t > T
- P
Without any loss of generality we have chosen q + 1 to be
even.
All terms in this expression have been previously de-
fined and we use the geometry of Fig. 2.
As an aside, we
mention that the particular form of the electric field used
here may need modification if thew= w condition is relaxed.
c
For example, if the cavity is tuned to the molecular frequency, w
= w , and then the carrier frequency is swept
across the cavity while monitoring the emission signal, one
can observe a modulation of the emission as the cavity selects
37
those Fourier components of the polarization pulse that fall
within its bandwidth at any particular frequency offset
w - wc.
The effect can be quite large for the short, square
pulses used here, with the amplitude of the emission signal
effectively tracing out the frequency envelope of the pulse,
completely disappearing when w - w_ is an exact multiple of
2TT/Tp.
The polarizing electric field envelope in the cavity as
described by Eq. (79) as a function of time is shown in
Fig. 3.
The finite rise- and fall-times encountered in this
high-Q system have been explicitly accounted for in the timedependent exponential terms in Eq. (79). Remembering from
Eq. (51) that the polarization is determined by the area
under the e(t) curve, and noting that areas A and A' in Fig. 3
are equal, we find that the finite rise and fall times do
not, to first order, produce any changes in the polarization
components compared to that produced by the square pulse also
shown in the figure.
Defining AN
= An N,
a
constant, and
using Eq. (52) one obtains
P±(r,t1) =
KftAN
~ - sin(Ke0T
-(x2+z2)/w 2
e
° sin ky) .
(80)
Considering this functional form, note that, whereas in the
waveguide (traveling wave) experiment the quantity P. (but
not the polarization itself of course) is nearly independent
of r, in the standing wave experiment P. may exhibit the
CD
Figure 3.
Electric field amplitude of the polarization pulse in the
cavity as a function of time.
The square input microwave
pulse of duration x with 10 ns rise and fall times is
P
distorted by the long time constant of the cavity.
Since
the molecular polarization depends only on the area of
these curves, region A here being equal in area to region
A', this distortion does not affect our results.
39
0)
E
IxJ - *
40
feature of spatial variations on a distance scale of less than
one wavelength of radiation.
scale of k"
If a characteristic distance
is assigned for changes in the amplitude of the
standing wave electric field along the cavity axis, then the
corresponding distance scale for changes in P. and AN along
that axis will be approximately (kice x ) ~ for KG. X
> 1.
Solutions to Eqns. (22-24) without the v • V terms are therefore strictly valid only for times x
such that a typical mole-
P
cule can move only a short distance along the cavity axis com-1
-1/2
pared to (k<e x„)
, or x
«
(kv<e )
' , which is expression
(54b) .
Using the result in Eq. (80) for P. in Eq. (76), evalu2
ating u
from Eqns. (69) and (75), and using the condition
w = w , one o b t a i n s
-(x2+z2)/wo2
E ( r , v , t ) = - 8TTQTe
°
s i n ky e
—=__ x
- ( t - t , ) / T 9 KftAn
•* —^—— cos(w ( t - t ^ + w t ^ x
/d r ' s i n ky' e
s i n {(<e x )
x
(81)
e x p ( - ( ( x - v x ( t - t 1 ) ) 2 + (z - v z ( t - t 1 ) ) 2 ) / w Q 2 ) x
sin k(y' - v ^ t - t ^ ) }
.
Modifications of this expression for the case where the mode
number q+1 is odd are straightforward.
Our concern right now
is with the shape of the emitted signal.
We temporarily drop
41
the time-independent terms in front of the integrals. Integrating the remaining expression over a Maxwell distribution
of velocities for the static gas, one obtains
-t/T 2
E(t)
"
e
cos(w t + wt,)
o
i
x
2
mv / 2 k T
/dvdvdv,
x y z e ""
/dxdydz sinky
e
2
° x
(82)
s m { (Ke0xp) exp(-((x-v x t) 2 + (z-vzt)2)/wQ2) sin k(y-v t) }
Here t - t , has been replaced by t, measuring now from the
end of the polarization pulse, and the primes inside the spatial integral have been dropped.
Molecular emission occurs
at the transition frequency w , modified by an envelope with a
T, exponential decay and a time and velocity dependent sixdimensional integral containing all the Doppler dephasmg
information, as determined from the movement of molecules
through the cell.
This general arrangement of terms is inde-
pendent of any approximations made in the expression for the
cavity normal mode.
Because of the simple form of the normal mode in Eq.
(75) used here, it is possible to continue to evaluate Eq.
(82) analytically.
Making a change of variables k(y - v t )
•*• u, the integrals over y and v are:
42
-1
k
w
~mvv / 2 k T
ff(q+D/2
/ dv„e
*
/
s i n ( u + kv t ) x
y
-oo y
-ir(q+D/2
-((x-v t)2+(z-vzt)2)/w
sin(<e T „ e
'
o p
Here we assume that
(¥}
(83)
2
sinu)du
1/2
t «
d
(84)
so that we may ignore edge effects due to the mirrors.
Using
sin(u + kv t) = sin u cos k v t + cos u sin kv t and the symy
y
y
metry properties of the two integrals one obtains
2
, oo
-mv /2kT
y
4k~ / dv e
cos kv t
x
o
(85)
1T(q+1,/2
2
2
2
-((x-vt) +(z-vt) )/w
/
sin u sin
(KGOT
e
p
o
sin u) du.
Using the fact that q+1 is even, the integral over u is a representation of the ordinary Bessel function of order one, J,.
This last expression becomes
,
2ir(q+l)k
2
-mv /2kT
oo
y
/ dv e
o
"
cos kv t
(86)
2
2
-((x-v t) +(z-v t) )/w
J
l
(Ke
T
o p
e
x
y
2
> •
7r(q+l)k
v
is just the cavity length d. The integral over
g
is standard (McGurk et al. Eq. (145)), giving
\
fc2 4s2
2ks / e- /
,
(87)
43
where s = U n 2 ) 1 / 2 Aw J"1,
D
and
^ D - £ (^82,1/2 ,
the Doppler half-width.
(88)
An entirely conventional Doppler-
broadened envelope is obtained.
Numerical studies carried
out with less approximate forms of the cavity normal mode Eq.
(75) and including the movement of molecules during the polarization pulse as in Eq. (51) showed no significant changes
from this result under the conditions of pulse lengths and
input powers normally used in our experiments.
It follows
from Eq. (87) that condition (84) may be written
„ (,2kT.l/2
,c. , m ,1/2 <^
,
< d , or
N _ )
v(—) (v.. .I,')
m
w 2kT'
'
2
(89)
^ << d ,
easily true for all cavity modes except possibly a fundamental
TEM
0
mode.
Implicit in Eq. (87) is the necessity of having
polarized and entire Doppler envelope of the transition of
interest, and this can indeed be shown to follow from the
conditions imposed in deriving that result.
(3kT/m) '
Choosing v_j.s =
for a Maxwell-Boltzmann gas, from Eq. (88) one
obtains
AW
D " IT
V
RMS "
kV
RMS *
(90)
Now if AwD < Aw, so that the carrier falls outside the Doppler
envelope, then condition (40) guarantees that the entire line
44
profile lies within Ke . However, if Aw D > Aw, so that the
carrier lies within the envelope, Eq. (54) gives
kV
K< X
RMS T p ~ A t V p
'
or
(91)
Aw D « l/xp ,
which, when combined with Aw < Aw D , ensures that the Fourier
components of the carrier cover the entire line profile, even
if Ke Q <_ AwD«
Using Eqns. (86) and (87) in Eq. (82), and replacing the
terms from Eq. (81), one has
2 2
KftAN
-p /w
E(r,t) = 16 Tr Q. —r-°- sin ky e
° x
~
-t/T 2 t 2/ 4 s 2
e
e
u -wrdvdv
{{
2H^ ff dvx^z
Ht
h
e
e
C0S
^
(W
t
2
)
o
(92)
+ wt,) x
i
/
M
2
2
; / dxdz
-(x2+z2)/w2
o x
e
O
- ( ( x - v x t ) 2 + (z-v z t) 2 )/w Q 2
J (K£
l
oTp
e
)}
'
still measuring from the end of the polarization pulse. If
Q_ is written as
«L-5f '
<93)
where a is the fraction of energy lost by a wave in one transit
of the cavity, then the effective length of the cavity may be
taken to be d
, typically 10 2 d for a Q of 10 at 10 GHz.
45
The four-dimensional integral in Eq. (92) contains two kinds
of information.
First, there is the slowly varying time
dependent envelope resulting from the overlap of the exponen2
2
2
2
tial x , z , and (x-vxt) , (z-vzt)
terms. Physically, this
describes the gain or loss in signal as molecules more through
the cavity beam waist.
Second, one finds that the concept of
a TT/2 pulse, generally obtained by adjusting ice x
maximum of J,,
to the first
is not directly applicable to the cavity
because of significant transverse variations in the electric
field amplitude.
Computer calculations carried out using more
exact forms of the cavity normal modes show no changes in the
lineshapes from those predicted by this analytically derived
result.
F.
Field coupling out of the Fabry-Perot cavity
Expression (92) is sufficient for analysis of lineshapes
seen at the detector, but does not of course represent the
actual field at the detector.
To estimate that quantity with-
out actually solving a complicated boundary value problem we
resort again to the standard methods of cavity circuit theory.
The energy stored in the electromagnetic fields in the cavity
after the molecules begin emitting is (in cgs units)
U = ^r / v (E2 + B 2 ) dV = i- / v E 2 dv ,
to be averaged over one cycle of the radiation fields.
refer to Eq. (92) for the field.
We drop the v„ and v_
(94)
We
46
dependence inside the J, term in Eq. (92) so that we may perform the integral over dv dv„. Substituting Eq. (92) into
x
Z
2
Eq. (94) and replacing cos (w t + wt,) by its average value
of 1/2, one obtains
~ /tcttAN \
2
l
^
,
,
-2t2/4s2
-2t/T~
x
™o I
(95)
2
(SI dxdz ^ ( K E Q T
2
2
2
-p /w
-p /w
e
°)e
° ) ** .
If the cavity is coupled so that Q = Q , = Q 2 , the power
reaching the detector will be
p
„„* = S- w~ •
out
Q_
o
Using Eq. (95) this is
p
out =
( 8 / 9 ) vQ
o
(96)
o
(dropping the terms in t)
w
o(l<al^Zlb>lANo)2lTWo
2d
x
(97)
2,
2
2 - 2
-p / w
1
( ^ - y / / dxdz J
irw/
(<e T_ e
° p
- p /w
° )
e
2
° P
or,
Ke
P
out
=
(8/9)
^ o
(
2
2
I <a I Mz I b> IANQ) TTWO d ( ^ \ - ) I
oTp
J (u) d u ) 2
(98)
o po
This may be compared to the expression for P . for a reflec5
tion cavity following a TT/2 pulse:
P
out = b
Q
ow o (|<a|uz|b>|AN0)2 V
.
(99)
47
Here V is the cavity volume.
The corresponding waveguide ex-
pression in the dominant mode far above cutoff is
P
out * b
(T)
% ( | < a | " z M A N o ) 2 Ji 2 (Ke0Tp) V ,
where % is the cell length, and V its volume.
(100)
In comparing the
waveguide experiment, Eq. (100), to the cavity experiment, Eq.
(98), it is useful to refer to Eq. (93).
Experimentally, we measure not the emitted power, but the
electric field, which may be obtained from Eq. (98) by using
the waveguide formula. One thus obtains a Q1/2
' dependence in
the signal from the cavity.
The presence of a radiation-
damping time constant directly proportional to Q ensures that
energy is conserved.
That is, the higher radiation energy
densities in the cavity shorten the radiative lifetimes of the
polarized molecules, causing them to release their energy to
the detector at a rate higher than they would in a waveguide,
and before collision and Doppler dephasing processes take over
32
and dissipate that energy in experimentally nonuseful ways.
Given a sufficiently high Q and number density of molecules,
this effect will result in an effective line broadening.
Since
we have no evidence that radiation damping is affecting our
lineshapes, we have omitted this effect in our theoretical
treatment.
Referring again to Eq. (98) we see that the measured
signal is proportional to the first power of the dipole moment
u, provided that one is working somewhere near the first
48
maximum of J,, this being characteristic of time-domain spectroscopy.
We also mention that the terms
Q_w , or (wd/ac) w ,
when combined with the w
dependence of AN for a gas described
2
at some temperature, lead to the w dependence in signal,
this also characteristic of the time domain experiment.
The presence of the functional form J, in Eqns. (98)
and (100) deserves some comment.
In the asymptotic limit
x •+ co
J, (x) •*• (-^r)1^2 (sinx + cosx)
1
,
(101)
1TX
resulting in a loss of signal for large K E T
the condition x
<< T„).
(subject still to
In the waveguide experiment Jid^e T J
replaces sin (ice x ) when the variation of the electric field
across the broad dimension of the waveguide is taken into
account.12 We should expect m general that sin(KG X ) will
be replaced by some function of KG; X
that is zero when
(<e x )=0, damped in the limit of large argument, and in
between displays a structure at least reminiscent of sine,
whenever a pure traveling wave electric field is replaced by
an electric field such that spatially separated molecules
experience electric fields that differ by more than just a
phase factor.
This is certainly the case in the cavity.
Physically this damping results from an increasingly severe
dephasing of adjacent dipoles as the total time integrated
electric field envelopes that they experience become different
by larger and larger amounts.
The dephasing is closely
49
related to the development of spatial variations in the polarization in the cavity on a distance scale much shorter than
k
as discussed in connection with Eq. (80). Experimentally
this effect is readily observable, both in the static gas
and in the pulsed nozzle molecular source, particularly with
33
large dipole moment molecules such as HFHCN
(5.6 debye),
where we can completely destroy the signal by increasing the
input microwave power above a very low level.
G.
Static Gas Experimental Results
In Sections E and F two results were derived for the
static gas experiment:
Eq. (92) for the emission envelope,
describing the product of an exponential e-t/T
' 2 relaxation
with a conventional Gaussian time decay in the short pulse,
power broadened polarization limit, and Eq. (98) for the
amplitude of the emitted signal as a function of the parameter
xeo x_,
again valid only in the short pulse limit described by
p
Eq. (54). In this section we examine the experimental results
as a function of t and Ke xDSome details of the experimental apparatus are shown in
Fig. 4.
Microwave power at frequency v from the phase stabi-
lized master oscillator (MO) is shaped into short, square
pulses of typical duration 1 us by PIN diode switch 1.
In the
lineshape measurements discussed in this section, and for most
molecules having an electric dipole moment of 1 debye or
greater, this pulse passes directly into the cavity.
For
Monitor
Detector
Pin Diode
SI
Pin Diode /Impedance
SI
/ Match
OJ
Pin Diode
S2
{
Variable
Attenuator
Circulator
Detector
Figure 4. Part of the experimental apparatus used to make
the static gas measurements reported in Section G.
o
51
small dipole moment molecules, and in the measurement discussed
in the next section, a TWT amplifier is used to obtain pulses
with peak powers of 2 Watts or more.
In this case switch
Si' is used to block the TWT noise during the detection sequence.
Switch 2 is used to protect the detector crystal from
the high
power pulse. The cavity input is critically coupled
by adjusting a waveguide tuner until the power reflected from
the input coupling is zero.
When the microwave energy enters
the cavity it creates a coherent, macroscopic polarization of
the gas as discussed in Section II.
The subsequent molecular
emission is detected by a balanced mixer with a 30 MHz i.f.
amplifier.
This signal band is mixed again with the 30 MHz
i.f. signal, amplified, digitized, and recorded.
In Fig. 5 we show the time domain signal from the OCS
J = 0-*-l transition at 12163 MHz.
Data was taken at a gas
pressure of less than 1 mtorr, with a polarization time x
2.5 us, and digitized at the rate of 0.5 us per point.
=
Digi-
tal points have been connected by straight lines in Fig. 4.
Switch S2 was closed at time t on the figure, 0.35 us after
s
the p o l a r i z a t i o n pulse had ended.
156.5 kHz.
The c a r r i e r o f f s e t i s Av =
A low-Q mode, Qr < 5000 was used in t h i s measure-
ment t o minimize r i n g i n g from the high power p u l s e .
Care was
taken t o o b t a i n an emission envelope free from a m p l i f i e r
filter
distortion.
From Eq.
(92) t h e envelope should be combination of
Gaussian and e x p o n e n t i a l decays.
If
or
in
to
Figure 5.
Transient emission signal obtained from the OCS J = 0 -> 1
transition at 12163 MHz taken in the TEMonr-7 cavity mode
with a static gas sample.
The switch S2 blocking the
detector was opened at time t_.
t' is the first point
at which the signal envelope can be measured.
The signal
was digitized at the rate of 0.5 ys/point and the points
connected by straight lines.
t
s
f
60
80
time {ftseconds)
in
to
54
,-*„ J i f ^ i ) 1 ' ' 2
(102)
is plotted as a function of t-t', where I is the envelope
amplitude, and t* occurs a short time after t_, one obtains
a curve with a straight line asymptote having slope (2s)
and intercept sT 2 ~
,
at t = t', where s is defined by Eq. (88).
This is shown in Fig. 6 choosing t' to be the first available
maximum in the signal.
We obtain s = 13.12 ys, compared to
an expected result of 13.7 ys for OCS at 293K, and a T 2 of
21
42 ys, corresponding to a pressure of about 1/2 mtorr.
Al-
though Eq. (92) has been derived subject to various restrictions
on pulse length and carrier offset, we have never observed any
significant deviations from this result over a wide range of
variations of these parameters.
We now take up the question of emitted signal as a function of the parameter KC x . We have already derived one result in Eq. (98) and have discussed the general features expected in this experiment.
The emission signal for the OCS
J = 0-*l transition as a function of the parameter <e x
was
measured for pulse lengths x
The
of 0.5, 0.8, and 1.0 ys.
XT
experimental apparatus is shown in Fig. 4.
A continuous flow
system employing a standard gas regulation valve was used to
introduce the gas into the cell.
The flow rate was adjusted
once at the beginning of the experiment and then monitored
during the measurements on a gauge capable of registering a
5% change in pressure at the pressure of one mtorr used here.
in
Figure 6. Plot of (-In
I(t - t')/I(t*)) ' versus t - t ' for the
signal in Fig. 4. Here I(t) is the signal envelope
amplitude. The solid line is (t - t'J/26.24 ys + 0.320,
and may be used to determine the Doppler and T 2 dephasing
times.
56
V
(
(.4)1
(.4-4)1
U|-)
57
No change was detected.
The cavity quality factor was deter-
mined by measuring the frequency position of the half-power
points of the transmitted power as the carrier was detuned
from the cavity.
For a half-power width of 1.38 MHz at 12163
MHz, one obtains a Q T of Q_ =
L
Ju
, ,» = 8814. A precision rotary
If J O
vane attenuator was used to adjust the input power to the TWT
amplifier.
Input and transmitted power levels were later
measured directly with
a power meter.
Equations (13) and (39)
were used to calculate e , where d = 57 cm, and Q , was set to
3QL«
We estimate this absolute electric field calculation to
be in error by less than 50 percent.
Experimental results are shown in Fig. 7. All measurements were taken with Av = 90 KHz. On
plot (p-jnDUt)1/2 T , which is measured
which is directly proportional to Ke x
maxima near 1.4 mW 1/2
' ys correspond to
the horizontal axis we
experimentally, and
. Ths pronounced
input powers of 5.5 mW,
2.3 mW, and 2.1 mW, for pulse lengths of 0.5 ys, 0.8 ys, and
1.0 ys, respectively, giving calculated values of K E T_ or
2.9, 3.0, and 3.6.
The signal is reduced by no more than a
factor of four from its maximum over a range of a factor of
sixteen in electric field, or about 24 dB in input power.
The short pulse result 34'35
/E0Tp
;
Ke
T
o p
°
1 - J 0 (K£ X )
J.
X (u) du =
Ke
°
°
oTp
p
(103)
in
oo
Figure 7.
Experimentally measured amplitude of the OCS J = 0 -*• 1
static gas emission signal as a function of the polarizing electric field for the TEM_n._ cavity mode.
Intensity (Arbitrary Units)
N
&
3'rS
3
5
co
l> D
ro
O)
6S
O
- o p
o oo o»
T:T=T=
</* tn tn
60
taken from Eq. (98) for the static gas signal as a function of
<eQx
is shown in Fig. 8.
at Ke0T
The peak value of 4.2 • 10
occurs
= 2.8. Although the width of the peak in the calcu-
lated curve is larger than the experimentally measured results,
overall agreement between the two curves is good. Using condition (54b) one would expect the short pulse expression Eq.
(103) to be valid for
xp «
(kv K e 0 )" 1/2
#
or
(104)
1
KE~T_ << KVT
°P
D
tr
Taking v = 3 * 104 cm s-1 for OCS at 293 K, this is KCoXp «
26 and 13 for pulse times x of 0.5 ys and 1.0 ys, respectively.
kr
H.
Comparison of Static Gas Fabry-Perot Results to
Waveguide Spectrometer Results
To conclude our discussion of the static gas, we use
Eq. (98) for the cavity experiment, and the corresponding waveguide experiment result in Eq. (100) to compare the strength
of the static-gas signal from these two types of time-domain
4
spectrometers. For the cavity, take Q = 3 ' 10 , <£_.?„ =
2.8, wo = 5.4 cm, and d = 47.5 cm for 12.2 GHz radiation; for
3
the waveguide, % = 6 m, V = 1400 cm
set J, to its maximum of 0.6.
obtains with these values:
for an X-band guide, and
From Eqns. (98) and (100) one
Figure 8.
Calculated amplitude of the static gas emission signal
as a function of KG x
mode.
for an arbitrary TEM00cr cavity
Intensity (Arbitrary Units)
Z9
63
P out (cavity) , / 2
<Pou^waveguide)) ' "
10
<105>
'
Any discussion of the ratio of signal-to-noise in the
two experiments must include several other considerations.
The cavity is obviously a narrow-banded instrument, and although this makes searches for new resonances more difficult,
it also reduces the noise power included in its normal working
bandwidth.
The bandwidth of the waveguide experiment however,
depends on a relationship similar to Eq. (4 0), and is generally
made as large as possible by using a high-power TWT amplifier
and wideband signal processing components, reaching values as
large as 50 MHz for molecules with transition moments on the
order of 1 debye.
In the Doppler broadened limit, from
Fig. 5, signal relaxation times will limit both experiments
to pulse repetition rates of perhaps 20 kHz or less.
Wall
relaxation, definitely not a problem for the cavity experiment,
may limit decay times in the waveguide cell to much less than
that shown in Fig. 5, particularly if a small cell cross
section is used to increase the microwave power density,
and thus the bandwidth.
In a carefully designed 20 foot
waveguide system the power-pulse ringtime can be made as
short as 200 ns, which, when combined with power pulse
lengths of 10 - 30 ns, make it possible to work well within
the pressure broadened regime without much loss in signal.
The larger number of molecules approximately compensate for
the shorter duration of the emission signal.
In this case
repetition rates of 300 kHz are routinely possible.
This
64
approximately even trade-off between the number density and
the signal duration changes if one is attempting to work with
weakly bound dimer systems in a static gas.
Here the signal
duration is still inversely proportioned to the first power
of the gas pressure, but the number of species of interest,
which we assume to depend on the product of the partial pressures of the two monomer components, can be made to be proportional to the square of the total pressure.
Having worked with both types of pulsed spectrometers,
we find that Eq. (105) probably overestimates the ratio of
sensitivities of these two instruments.
The two spectrometers
are sufficiently different that a choice for a static gas
experiment should be heavily weighted by the other listed
considerations.
Of course, the tremendous increases in sensi-
tivity and resolution of the Fabry-Perot instrument come when
the large volume of the cavity is combined with the pulsed
nozzle molecular source, and it is this combination that is
taken up in the following chapter.
65
Chapter III.
A.
The Gas Dynamics of a Pulsed Supersonic
Nozzle Molecular Source as Observed with
the Fabry-Perot Cavity Microwave Spectrometer
Introduction
Having developed the theory of pulsed Fourier transform
spectroscopy carried out in a Fabry-Perot cavity in the previous chapter, and having treated many of the details of this
type of spectroscopy in the static gas discussion, our emphasis in this work will be on the properties of the gas nozzle
expansion itself.
In Part B, we review the derivations and
results from the previous chapter that are applicable to arbitrary distributions of molecules in the cavity.
In Part C,
these general results are used to obtain the equations describing the emission lineshapes produced specifically by a
gas m
the nozzle expansion.
The characteristics of experi-
mentally observed lineshapes are analyzed according to these
results.
From this we obtain information about the speeds,
density distributions, and dephasing properties of molecules
in the expansion.
All work done in this section is with the
nozzle positioned at its normal location midway between the
mirrors, pointed directly at the center of the cavity.
In
Part D we discuss nozzle configurations in which the nozzle
is either tilted or relocated from its normal position.
These
experiments provide additional information about the beam, but
may also be of use in choosing a geometry for the nozzle and
mirror system.
66
B.
Equations Describing the Polarization and Emission of
Radiation by Molecules in the Fabry-Perot Cavity
Appropriate for the Pulsed Nozzle Problem
The theory describing the absorption and emission of rad-
iation by molecules in the standing wave electric field of the
Fabry-Perot cavity has been developed in detail in the previous
chapter.
We summarize here the results necessary for the pres-
ent study.
When a number of non-degenerate two-level quantum
systems are brought into contact with a standing wave electric
field E(r,t) = 2 z e(r,t) cos wt (we use the geometry of Fig.
**
fV
***
f*
9) it can create a macroscopic polarization, P, of the gas
of the form
P(r,v,t) = (P„(r,v,t) + i P. (r,v,t) )ela)t + c.c. ,
where P
(106)
and P. are real-valued functions of the six phase-space
coordinates r, v, and of t, and 'c.c.' indicates complex conjugate.
The polarization components P
and P., along with the
two-level population difference per unit volume associated with
the velocity v, AN(r,v,t), satisfy the Bloch-type coupled
partial differential equations:
(ft + Y • V)P r + A ^
+ ^ =
0 ,
(ft + v • V)P ± - Ao>Pr + K 2 e»'r,t) ^
( |_
+ ^- = 0 ,
(107)
*AN
, _
, *(AN-ANQ)
+ v . V) ^
- e(r,t)P. +
= 0 ,
4Ti -
where V is the gradient operator, A<D = to -w, the difference
between the molecular transition angular frequency w , and the
6?
Figure 9.
Geometry and coordinate system used for the gas nozzle
and Fabry-Perot cavity.
The density of the gas expan-
sion is parameterized as (cosp 0)/r .
The microwave
radiation travels along the y-axis, perpendicular to
the nozzle axis.
68
carrier angular frequency y, T-, and T 2 are phenomenologically
introduced first-order relaxation terms, AN
is the equilibrium
value of AN in the absence of an electric field, and
K
= 2ft"1|<a|yz|b>| ,
(108)
where <a|yz|b> is the electric dipole transition moment connecting the two levels.
When the gas is introduced into the
cavity via a pulsed nozzle it is possible to assign to each
coordinate r in the cavity an unique velocity v(r).
It is con-
venient to define new quantities p (r,t), p.(r,t), and
An(r,t), by
N(r,t)pr(r,t) = / dv Pr(r,v,t)
N(r,t)p (r,t) = / dv P.(r,v,t)
(109)
N(r,t)An(r,t) = / dv AN(r,v,t).
Here N(r,t) is the number of two-level systems per unit volume.
We will assume that the electric field interaction does not
affect the center-of-mass motion of the molecules.
We will
also assume that N(r,t) satisfies the conservation equation
3N(r,t)
+ v • (v(r)N(r,t)) = 0 .
3t
The quantitites p , p., and An, then satisfy
p
(
(
ft
+
1
)p
+
Y -* ' ! r A w P i + f
£ = 0
ft
+
Y(5> * !>?! ~ Aw P r +
e(r,t) *|B. + _i = o,
2
K
(110>
'
P<
(111)
69
a
(
*Ar,
+
( )
}
•
e(
-nfAn-AnJ
t)p +
ft Y 5 ' ! ^ T ~ 5' i — 4 T — ~
where An
=
°'
(112)
may be taken to be a constant.
The Fabry-Perot cavity is inherently a narrow-bandwidth,
high electric field instrument, so we concentrate on solving
the on-resonance, Aw = 0 form of these equations. When the
polarization pulse length x
is sufficiently short that a
kr
typical molecule travels only a short distance compared to the
characteristic distance of variations in p , p., and An, and
provided that x_ >> T,,T2, we may ignore the v • V and T^,
T 2 terms in Eqns. (110-112).
The solutions to the resulting
equations (with Aw set to zero) are
pr(r,t) = 0,
(113)
K&An
j-S. sin (</ e(r,t')dt'),
p. (r,t) =
An(r,t) = An
t
cos (K /
e(r,t')dt') .
%
(114)
(115)
~
The conditions necessary for these solutions to be strictly
mathematically valid are:
2
2
i. (K£Q) >> Aw ,
ii.
iii.
iv.
x
/* 2 ,
<< xe /Aw
x
-1/2
<< (kvice ) ' ,
(KE 0 )
2
>> T 2
-2
, x
(116)
«
T2 ,
70
where z
is one-half the maximum electric field amplitude in the
cavity, and v is a typical molecular velocity component along
the cavity axis.
To obtain an appropriate form for e(r,t') in expressions
(114) and (115), we note that the frequency spacing between
successive nondegenerate modes in the Fabry-Perot cavity is
much greater than the width of any single mode. We will work
in this chapter exclusively with the TEM
modes. Experi-
mentally, we find no significant variation in lineshapes taken
in different cavity modes, and the approach here using TEM
modes may easily be generalized. The electric field e(r,t')
is separable as e(r)f(t'), where, for the TEM
£{
V
with
p
= £
w
2, 2 , .
e P
o wlyT "
cos(ky-7rq/2) ,
= x + z
/STT
(117)
(see Fig. 9), and
w^ = (^ (d(2R -d)) 1 / 2
O
mode:
O
1/2
)
,
(118)
W(5 =
" "° (1 + fr) :
w
c ,
k = -2
c '
\ is the free-space radiation wavelength, d is the mirror
separation, R the radius of curvature of either mirror, and
q + 1 the number of half-wavelengths between the mirrors. The
beam waist, w , is the distance from the center of the cavity
to the 1/e points of the field strength.
The normal mode from
Eq. (117) has been simplified from the more exact expression
(Eq. (73) in chapter II by neglecting certain terms in the
cosine argument.
To be consistent, we will use in this
cahpter a simplified equation
to relate the frequency and mode number to the mirror separation.
The polarization pulse inside the cavity is assumed to
begin at time t' = 0 and end at t' = t-.. As explained in
chapter II, although the narrow-band cavity may severely distort the shape of the square microwave pulse of duration x
P
incident upon it, the pulse area remains unchanged, so the
integral in Eq. (114) may be immediately done to obtain, at
the end of the polarization pulse inside the cavity:
p
(
t
i £' l
) =
K*iAn_
w
4~^ sinl<e o T p |^) e
2 .2
cos (ky-Trq/2)}, (119)
P /W
with a similar result for An.
After the polarizing radiation has been removed, p ,
p , and
n continue to evolve as described by Eqns. (110-112)
without the e(r,t) terms.
The polarized gas now produces an
electric field in the cavity.
Using exact solutions to Eqns.
(110-112) without the e(r,t) terms, and using Maxwell's equations to couple the polarization to the emitted electric
field, one obtains for the emitted electric field in the
cavity:
72
2,
E ( r , t ) - 8TTQ_ ( W » e "
/w
p
2
cos(ky-irq/2) x
e" t / / T 2 c o s UOt + ( w - w
j t , ) — V2-,
O 1
7TW
x
d
(120)
3
/ d r« p (r'-v(r')t,t )(w/w(y'))
V
1 ~
~ ~
<
p
N(r',t) e"
2
/
/w
2
X
.
x
o
I,
lY ;
cos(ky'-7rq/2) .
This has been simplified from the more general result Eq. (76)
in chapter II by setting the cavity resonance angular frequency
w c equal to w . Time is measured from the end of the polarization pulse at t,.
The quantity p.(r'-v(r')t,t,) is to be ob-
i
i —
— ~
i
tained from Eq. (119) .
As discussed in detail in chapter II, Eq. (120) has been
derived under the assumption that the velocity field v(r)
satisfies
v(r) * V v, = 0 , i = x,y,z ,
(121)
which may be verified directly for the pulsed nozzle velocity
field to be given below.
Physically, we require that all
molecules travel at constant speeds along straight line paths.
The normal mode form (w /w) e"p ' w
cos (ky-irq/2) outside
the integral in Eq. (120) simply describes amplitude variations
in the emitted electric field signal characteristic of the
TEM
mode.
The essential lineshape information is contained
in the time-dependent e~ ' 2 cos(wQt + (w-w )t.) terms, and
73
in the spatial integral, which contains all information related
to the molecular motion, including Doppler dephasing, and
dephasing due to the movement of the molecules through the
cavity. To proceed further we need to consider the exact
form of the pulsed-nozzle molecular velocity field.
C.
Lineshapes in the Pulsed-Nozzle Fabry-Perot Experiment
To complete the derivation of the functional form of the
time-domain lineshapes seen in the pulsed-nozzle experiment
we consider certain characteristics of the pulsed nozzle gas
expansion.
These may be summarized as follows:
1. All molecules travel at constant speed v
on radial
paths originating at the nozzle. Using the geometry of Fig. 9,
the velocity field v(r) may be written
xx + yy - (h-z)z
v(r) = v - ^
^
YT72 '
(x^ + y* +
(122)
{h-z)*)*-'*
which satisfies Eq. (121).
2.
To begin our analysis of the gas expansion we will
assume a molecular density distribution N(r,t) of the form
H(r,t> c
go4i=
2
r
(h-s)P
.ov2J.„2a.,._ .2,p/2+l '
(x'+y"+(h-z)')
(123)
where r and 6 are the radius and polar angle from the nozzle.
The exponent p will be determined experimentally.
3. As has already been noted, in the expanding gas
T 2 > 100 ysec, justifying our neglect of relaxation processes
74
during the polarization.
Using Eqns. (119), (120), (122), and
(123), and dropping the normal mode form and numerical constants from in
front of the integral in Eq. (120), one ob-
tains for the emitted electric field lineshape:
E(t) = e""t/,T2 cos(w0t + (w-wo)t1)
'v
x
d 3 r sin k y s i n ( K e T (
o p w<y-5 t)>
x
(124)
2
exp(
2
-(x-v t) -(z-v t )
s
) sin k(y-v t) } x
y
w (y-v t)
, Wo x
,-x -z . cos p 0
(
) exp(—^
*
2—
w(y)
w (y)
r
where v , v , and v
'
are functions of x, y, and z.
We have
chosen q + 1 to be even, and have dropped the primes inside
the integral.
Note that the integrand contains two gaussian
waist terms, one inside the sine expression and time dependent,
and one outside the sine and time independent.
Physically
these describe the loss of signal as the molecules move out
of the beam waist.
Because of the complex algebraic form of Eq. (124) it
is not possible to simplify this result analytically.
It
is useful, however, to consider two special cases that can
be treated analytically, these being tightly collimated beams
moving at uniform speed v Q either perpendicular or parallel
to the cavity axis.
In the perpendicular case
v x = v y = 0, v z = v Q ,
75
and the number density N(r,t) is
N(r,t) = N 6(x)5 (y) ,
(125)
where 6 is Dirac's delta function.
Using Eq. (124) (with
q+1 odd in this case) and Eq. (125) one has
-t/T 2
E(t) = e
cos(wQt + (w-w0)tx)
/"sin {Ke0x
/z-v t\ 2
/z_\ 2
e " ! - ^ - ) } e " U 0 ) Az ,
(126)
which is a single resonance centered at w , decaying with an
envelope combining T 2 dephasing with movement of the molecules
out of the cavity.
In the parallel case
vx = v z = 0.
' vy — v o', and
N(r,t) = N Q 6(x)S(z) ,
(127)
for -d/2 + v t <_ y <_ d/2, and zero otherwise. Using Eqns. (124)
and (127) with v > 0, one has
o
_t T
E(t) = e / 2 cos(wQt + (w-wQ)t1)
d/2
/
dy sin ky sin{xe_x_ sin k(y-v t)} ,
(128)
-d/2+v t
o p
o
o
dropping the w(y) dependences in Eq. (124) . Making a change
of variables ky - kvQt ->- u, the integral is
k-l;7r(q+l)/2-kvot d u s i n ( u + k v t ) B i n ( K e
-TT(q+l)/2
°
- S in u ) . (129)
° p
76
The -kvQt term in the upper limit is of minor interest here,
so we ignore it for v Q t << d.
Expanding sin (u + kv t) and
remembering that q + 1 is even, one obtains a lineshape
-t/T2
E(t) = e
cos(wQt + (w-wQ)t1)d
x
(130)
cos
( w o "ft) J l< Ke oV '
where we have used (q+1)(X/2) = d, and J, is the ordinary Bessel function of order one.
The resulting line E(t) consists
of the two frequencies wQ(l+v /c)
and wQ(l-v / c ) , with an
obvious physical interpretation.
By including the -kv t term
in the upper limit of Eq. (129) , it can be shown that there
will be no Doppler splitting in the TEM
mode.
We return now to the full pulsed nozzle expression Eq.
(124), and consider the properties of this result in comparison to experimentally observed lineshapes.
For the purpose of
introducing the unusual features of these time-domain spectra
we refer to Figs. 10 and 6.
Figure 10 is the time domain
16 12 32
record for the J=0 ->• 1 transition of
O C S , which is
known to consist of a single line at 12163 MHz.
The spectrum
was recorded by pulsing a 4% mixture of OCS in Ar through the
nozzle.
We use a thin plate flat orifice bolted to the bottom
of the pulsed valve; we do not use a skimmer.
The signal was
digitized at the rate of 0.5 ysec/point and the points connected by straight lines.
The corresponding power spectrum is
shown in Fig. 11 and has a frequency resolution of 3.90625 kHz/
point.
The most prominent characteristic of this spectrum is
Figure 10. Time domain signal from the
O C
S J = 0 -*- 1 transi-
tion, converted at 0.5 sec/point. The points have been
connected by straight lines. Gas mixture is 4% OCS in Ar.
78
o
CM
o
CO
o
<v
tn
8
o
8
Ld
vo
Figure 11. Power spectrum obtained by adding 256 zeros to the data
in Fig. 10 and Fourier transforming. The resolution is
3.90625 kHz/point, with a splitting of 36.3 kHz.
—»•»—••••••••
FREQUENCY
00
o
81
the symmetric splitting apparent in both figures.
This split-
ting is a general feature of all spectra taken with the pulsed
nozzle in the Fabry-Perot cavity.
We have found that the
position of the center of this pattern is invariant under
all changes in spectrometer operating conditions, and corresponds to the molecular resonance frequency.
The lineshape
itself provides several different kinds of information about
the gas expansion, and will be our main concern in the remainder of this paper.
It is convenient to separate Eq. (124) into two parts.
The cos(w t + (w-w ) t.,)exp(-t/T2) terms center the emission
signal at w
and provide a routine exponential damping.
The
interesting lineshape information is contained in the remaining integral expression, which we denote I(t) . This envelope
function I(t) contains two types of signal damping.
As dis-
cussed in the special case of a narrow beam traveling through
the cavity perpendicular to the axis, there is a fall-off in
signal due to the transverse motion of the molecules out of
the cavity, mathematically expressed as the overlap of the
2
2
2
2
(x-v t) , (z-v t) and x , z terms in the exponentials. A
x
z
second type of damping, Doppler dephasing, results from the
movement of molecules from the region where they were polarized with one phase, to regions where they would have been
polarized with a different phase, mathematically appearing
in the k(y-v t) argument.
Numerical studies show that in the
pulsed beam, the damping due to transit of the molecules out
82
of the cavity is negligible in comparison to the Doppler dephasing.
Because inclusion of this transit-time damping in our
analysis would provide at best only a marginal improvement in
our description of the gas expansion, we will not include this
effect in our analysis. Accordingly, we will use here the
lineshape expression
w
3
J
I(t) = /'cavity
.. d r "
sin
sin {ice
°
" ky
""•"
^ o 'x,
p w(y)
exp
exp
3111k v t > }
"HrM
^- y * wTyT
\ w (y) /
x
(131)
cos p 8
r2
\i/(y) /
In Fig. 12, we show the triple integral Eq. (131) calculated for the TEM Q Q 3 1 cavity mode as a function of t, for
p = 0.5, corresponding to a cos0 "56 density distribution,
a frequency v
cm/sec.
= 10 GHz, and a molecular speed v Q = 4-10
No relaxation processes except Doppler dephasing are
included in this calculation.
The time, L , from the maximum
of the curve at t = 0 to the first zero crossing, and the
distance L-, i = 1, 2, ..., between successive zero crossings
are,
in microseconds, 41.8, 47.0, 46.6, 46.4, and 46.5,
respectively.
These spacings conform to a general pattern,
independent of any of the parameters in Eq. (131), in which
the spacings L. following the first zero are nearly constant,
00
Figure 12. The envelope function Eq. (131) calculated for a density
0 5
2
distribution (cos " 6)/r , v = 1 0 GHz, molecular speed
4
v = 4-10 cm/sec, and a mirror spacing of 48 cm. No
T~ is included in this calculation.
84
but possibly quite different from the f i r s t spacing, L . The
r a t i o L Q / L i i s an important parameter, independent, as we
will show, of v and
p.
v , depending only on the coefficient
I t varies monotonically from 0.85 for an isotropic dis-
tribution with p = 0, to 1.07 for p = 2, t o a r b i t r a r i l y large
values as p increases.
As p increases t h e f i r s t zero crossing
moves away from the origin, and the heights of the successive
maxima in the curve become smaller in r e l a t i o n to t h e height
of the curve at t = 0.
A calculation for p = 4, where a l l
other parameters are held fixed from the calculation in Fig.
12 is shown in Fig. 13.
In the limit of l a r g e p then, the
envelope will show no o s c i l l a t i o n s , as i s required for a
t i g h t l y collimated beam traveling perpendicular to t h e cavity
axis, as has already been discussed.
Figures 12 and 13 i l l u s t r a t e the very important result
t h a t the spacings L-, i > 0, are independent of p.
This r e -
s u l t is useful in the present analysis because i t w i l l permit
us to consider the velocity and frequency dependence of the
Doppler s p l i t t i n g independently from the s p a t i a l d i s t r i b u t i o n
of the gas expansion.
In the frequency domain t h i s inde-
pendence i s not as clear-cut, since the i n i t i a l part of the
time domain spectrum up to the f i r s t zero crossing may change
the frequency s p l i t t i n g , or even eliminate i t altogether, as
in the case of large p .
Calculations show that the frequency
s p l i t t i n g in the power spectrum agrees to within 10% with the
inverse average zero spacing 1/L of the time domain of p < 2.
To f i r s t order for p < 2 the i n i t i a l time domain lobe only
00
Figure 13. The envelope function Eq. (131) calculated for a density
4 0
2
distribution (cos ' 0)/r and otherwise the same parameters as Fig. 12.
87
O
Ixl
2
88
fills in the center of the power spectrum pattern.
For strong,
well resolved spectra, this splitting invariance is unimportant, since the peak midpoint position is always invariant
under all changes in spectrometer conditions.
For very weak,
complex spectra, however, where it may be difficult to deconvolute the Doppler splitting except after taking many spectra,
during which time the nozzle source conditions may change,
it is important.
In Fig. 14 we show the power spectrum of the curve in
Fig. 12.
The envelope in Fig. 12 was multiplied by an arbi-
trary cos wt term before it was transformed, to displace the
power spectrum along the frequency axis.
The peaks in Fig.
14 are separated by 2 x 10.23 kHz, which may be compared to
the value 2 w o / c of 2 x 13.33 kHz for v = 10 GHz, and v =
4*10
cm/sec.
The splitting may be understood intuitively as follows.
Consider the electric field emitted by a single narrow beam
traveling on a radial path centered in the zy-plane, away
from the nozzle at angle 0 with respect to the z-axis.
lineshape
The
in this case may be obtained from Eq. (131) by
dropping the cosp9 term in that expression and restricting
the angular integration to the differential solid angle occupied by the beam.
The lineshape for the special case of
0 = 0 ° has already been given in Eq. (126).
When 0 = 90° we
obtain a result similar to Eq. (130). Numerical studies show
that for arbitrary angle 6 the lineshape is closely
00
Figure 14. Power spectrum obtained by multiplying the envelope in
Fig. 12 by an arbitrary term cos w^t and Fourier transforming.
The peak separation is 20.46 kHz.
FREQUENCY
O
91
approximated by the expression
vQsin6
t J
E(t,0) cc cos(wQt + (w-wQ)t1) cos(w Q —
) i('ce0Tp^ i
where the v
(132)
term in Eq. (130) has been replaced by the velo-
city projection v sin0 onto the cavity axis.
Here we omit
the T 2 dependence, and any loss of signal that occurs as
molecules move out of the cell.
Expression (132) is valid
for values of the parameter K E T
<_ 4 and is not affected by
removal of the beam waist terms in
Eq. (131) .
In a time
domain experiment we excite and detect the emission from all
molecules in the cell simultaneously.
The total emitted field
is obtained by adding contributions from all 0, weighted by the
angular density distribution.
We continue to deal with the
problem where all molecules lie near the zy-plane, since inclusion of an azimuthal angle makes the subsequent algebra
impossible to carry through analytically.
One then obtains
for the total emitted electric field
E ( t ) <* dJ^KCQXp)
C O s ( w o t + (w-W Q )t ; L ) X
//2coB(a»0V°
0
= d(j)
J-^KEQX
IP" 1 ? 11
p/2
( ^
*)
S
^n
9
t) cosp 0 d 0
) cos(wQt + (u-w0)t1)
J D / ? U n ^ t)
p / 2
°
c
.
(133)
x
(134)
92
Note t h a t t h e p a r a m e t e r KE-T a f f e c t s o n l y t h e a m p l i t u d e ,
t h e s h a p e , of t h i s r e s u l t .
The c h a r a c t e r i s t i c
not
functional
form
J
(X)
D
P // ^2
xP/2
(135}
U 3 5 )
containing a l l the information r e l a t i n g t h e s p a t i a l
t i o n of m o l e c u l e s i n t h e e x p a n s i o n t o t h e o b s e r v e d
distribulineshape
i s i n e x c e l l e n t q u a l i t a t i v e agreement w i t h t h e more e x a c t ,
3-dimensional numerical r e s u l t already discussed.
We a l s o
mention t h a t t h e c u r v e i n F i g . 12 b e a r s a c l o s e r e s e m b l a n c e
t o ( s i n x ) / x , t h e F o u r i e r t r a n s f o r m of a s q u a r e a b s o r p t i o n
curve in frequency
space.
We h a v e s t u d i e d t h e b e h a v i o r of t h i s f r e q u e n c y
separation
Av as a f u n c t i o n of t h e gas m i x t u r e s and of t h e f r e q u e n c y of
the molecular t r a n s i t i o n .
I n F i g . 15 and Table 1 we show
r e s u l t s f o r t h e s p l i t t i n g of t h e J = 0 •> 1 l i n e of OCS a t
12163 MHz i n v a r i o u s m i x t u r e s o f He, Ar, and Kr.
The m o l e -
c u l a r s p e e d v for a g i v e n m i x t u r e should b e d i r e c t l y
-1/2
tional to m
propor-
' , where m is the concentration weighted mole-
cular weight of the mixture. The observed peak separations
-1/2
for the four different gas mixtures follow the Av = m '
rule quite closely.
This behavior can be determined from
the integral expression in Eq. (124) by inspection, since v Q
and t never occur in the integral in Eq. (124) except as the
product v t.
93
i
1
i
1
T
100
d/
N
V
X
50
<
o7
0,0
i
1
1
1
1
0.1
0,2
0.3
0.4
0.5
/*/m
Figure 15.
Splitting of the OCS J = 0 -*• 1 transition at 12163
MHz as a function of the inverse square root of
the concentration weighted molecular weight of the
gas mixture for four different mixtures.
splitting values are given in Table 1.
The
Table 1.
Splitting of the OCS J = 0 -> 1 transition at
12163 MHz as a function of gas mixture.7
Mixture
Av (kHz)
a.
96% Kr, 4% OCS
27.7 + 2
b.
96% Ar, 4% OCS
35.2 + 1
c.
71% He, 25% Ar, 4% OCS
60.5 + 2
d.
96% He, 4% OCS
89.5 + 2
mirror separation 47.3 cm.
95
The complementary experiment is to study the splitting
as a function of frequency for a given molecule in a constant
gas mixture.
Some experimental results are shown in Fig. 16.
We find that Av is directly proportional to v.
lated from Eq. (131) are listed in Table 2.
Results calcu-
In addition to
adjusting the frequency in this computation, it was necessary
to adjust the mode number q so that the mirror separation
remained approximately constant. As the mirrors are brought
closer together, with the nozzle remaining positioned at their
midpoint, they constrain the geometry of the expanding gas in
such a way that the splitting becomes dependent on that separation, falling to zero as the TEM Q 0 0 mode is approached,
and asymptotically approaching a constant value in the limit
of large separation.
Because of the direct proportionality between Av and
v , and between Av and v, and because the frequency splitting
is nearly independent of p, as has been noted above, we may
use the Av values from Table 1 to determine the speed v of
OCS in the various rare gas carriers. From our discussion
of Fig. 14, the relationship is
„
V
_ Ave 13.33
n
(136)
o " T V 10723 '
from a 48 cm mirror separation.
,M
Values for speed v
obtained
in this manner are listed in Table 3. For comparison we also
list values for the terminal speed v_, obtained from the
. 23
expression
96
N
X
<
Figure 16.
Doppler splitting as a function of frequency for
several different molecules.
Gas mixture in each
case was 4% halogen ocid in the corresponding rare
gas carrier.
97
Table 2
Calculated frequency splitting Av as a function of v.
number
Frequency v
(GHz)
Av
(kHz)
Av/v
d
(cm)
17
6
12.45
2.08
45.0
18
6
13.30
2.22
47.5
24
8
16.95
2.12
46.9
25
8
17.60
2.20
48.8
30
10
22.00
2.20
46.5
31
10
21.75
2.18
48.0
37
12
26.15
2.18
47.5
44
14
29.75
2.13
48.2
<3
a
Calculated for v Q = 4*10
4
cm/sec.
98
Table 3
Speed of OCS at 4% concentration in
various carrier gases.
Carrier Gas
v„, (cal'd)a
,*,«4
(xlO cm/sec)
Speed (expt'l)
_/_x
(xlO,4 cm/sec)
He
14.38 + 1.5
13.9
Ar
5.66+0.6
5.4
Kr
4.45 + 0.5
3.8
r = fe_ kT\ 1/2 , where Y =
T I^Y-1 m J
T = 293K.
c
p/
c
v
= 5/3, a n d
99
V
(
T
Y -1
"m}
(137)
,
where v-, is the speed far from the nozzle that would be attained in the limit of zero temperature of the expanding gas.
Here T is the gas temperature before it reaches the nozzle,
andy = C p / C v
= 5
/ 3 . Values for v Q and v
are nearly identi-
cal, indicating that the OCS molecules apparently travel at
the same speed as the rare gas atoms.
Systematic errors in
this calculation include neglect of transit through the cavity,
the simplified cavity
normal mode form, and small differences
in the apparent splitting as measured in the time and frequency
domains due to the initial lobe in the time domain envelope.
Having established that Eq. (131) is capable of explaining certain general properties of the experimentally observed
lineshapes, we now analyze some specific time domain spectra.
The following data was taken in the TEM Q0 __ cavity mode at a
Q of 6300.
A four-percent mixture of OCS in Ar at room temp-
erature and at a source pressure of between 2.13 atm. and
2.07 atm. was expanded through a 0.020" diameter nozzle
opening.
Input microwave power was 1.5 mW.
Using the cavity
23
equation
E
=1
128 R Q_2 \ 1/2
%-
,
(138)
• (
2
when R = 1.5 mW, Q L = 6300, V = 12163 MHZ, Q c l = 3 Q^, W Q
2
-3
30 cm , and d = 46 cm, one has E Q = 5.9*10
esu, and eQ =
100
—3
fi
—1
E Q /2 = 3*10
esu. This gives KBQ - 2.8*10 sec . Data was
taken at a frequency offset Av = 182 kHz, giving Aw = 1.1*106
sec
, and
2
=0.167 «
(£)-
1.
(139)
For the 0.5 usee pulse time used here, <eQT
4
v = 6*10
= 1.4, and using
-1/2
cm/sec from Table 2 for Ar, (kvice )
= 1.52 ys.
We are therefore operating strictly within the boundaries set
by Eq. (116).
The pulsed valve is opened by a square voltage pulse that
operates a solenoid mechanism.
We define the delay time A
as
the interval from the time the leading edge of this voltage
pulse reaches the valve, to the time that microwave power is
applied to polarize the gas.
A plot of observed signal-to-
noise for the ArHCl van der Waals complex as a function of
the delay A, is shown in Fig. 17. After a short interval,
during which the valve opens and molecules travel to the
center of the cavity, the signal rises sharply, then trails
off slowly after the main group of molecules has passed
through the cavity.
From this graph it is apparent that for
short times after molecules reach the cavity, the properties
of the expansion are probably undergoing major changes on a
timescale of milliseconds or less.
The spectra are shown in Figs. 18-21.
are connected by straight lines.
Digital points
Digitization time is 128
I-"
o
Figure 17. Observed signal from an ArH
CI J = 2 •>• 3 quadrupole
hyperfine resonance at 10069 MHz as a function of the
delay A. between the leading edge of the square voltage
pulse that opens the nozzle value, and the microwave
polarization pulse.
SIGNAL (arbitrary)
o
m
5
Cfl
CD
O
30T
o
Figure 18. OCS J = 0 -*• 1 signal taken at a delay A
of 3.6 msec. A
4% mixture of OCS in Ar was expanded through a 0.020"
diameter flat plate orifice. The superimposed curve is
calculated from the envelope function Eq. (131), multiplied by an exponential e~ ' 2. For A = 3.6 msec we
find COS-0 '50 and T 2 = 160 ysec. The signals shown in
Figs. 19-21 were taken under identical conditions to
this one, except at different delays.
10-+
105
106
107
ysec/point.
The four spectra were taken at delay times A
3.6 msec, 4.2 msec, 4.5 msec, and 5.0 msec.
of
Lineshape dis-
tortion was minimized by using a low-Q mode to eliminate ringing, and by using wideband signal processing components.
Superimposed on each signal trace is a curve calculated from
Eq. (131).
Each fit has three adjustable parameters:
p,
which is determined solely by the ratio L /L, , v , and T 2 <
In all cases we obtain
v
~6*10
cm/sec, except Fig. 21
which was arbitrarily fit using this value.
We find that
the molecular distribution in the pulsed beam, as characterized by the single parameter p, varies from -0.5 at A. = 3.5
ms, to p >_ 3 in the later stages of the expansion.
A negative
exponent indicates a depletion of signal from the axis of the
beam.
We measure only the population difference density AN
between the OCS J = 0 and J = 1 rotational levels, of course,
and not the particle density.
Since the molecular distribu-
tion appears in Eq. (131) only in the integrand, the observed
lineshapes cannot be regarded as an extremely precise test
of that distribution.
Nevertheless, the calculated curves
do reproduce the data quite well, and the general trend in
the distribution from on-axis depletion at early times, to
high directivity at large times is well established.
The
values for T 2 obtained here must be regarded as a lower
bound for this parameter because of the omission of the
transverse motion terms in going from Eq. (124) to Eq. (13]).
Note that T 2 remains nearly constant for all four spectra
despite the severe damping apparent in the late spectra.
109
D.
Lineshapes Taken with Alternate Nozzle Geometries
Up to now all discussions of the pulsed beam experi-
ments refer to the standard spectrometer configuration with
the gas nozzle positioned at z=h (see Fig. 9 ) , exactly midway
between the mirrors and pointed directly at the center of the
cavity.
We now consider experiments in which the position
and orientation of the nozzle is changed.
The first set of experiments involve a rotation of the
nozzle in the zy-plane through an angle 9
with respect to the
(-z) axis, keeping the nozzle opening fixed in space at (0,0,h),
as shown in Fig. 22. Signals were recorded for the J = 0 -»• 1
transition in OCS at angles ranging from 8
= 0°, the standard
configuration, to 90°, for three different rare gas carriers.
Source and nozzle conditions were 2.4 atm of 4% OCS at 293K
expanded through the 0.020 inch diameter flat plate orifice.
Results are summarized in Table 4.
Referring to the
top row of figures for each gas, note that the Doppler splitting is virtually independent of nozzle orientation for all
angles 0° —
< 0o —
< 90°. We have made modifications to the
density term in Eq. (29) appropriate for the tipped nozzle
experiment and find that the calculated Doppler splitting for
distributions cosp(0-0 ) up to at least p - 2 vary by less
than 10% in this range 0° <- 0
<_ 90°. The calculations are
therefore consistent with the experimental results, which
were taken with short delay times A*. t o keep the Doppler
peaks well resolved.
For discussing the lower set of
Geometry and coordinates used for the
tipped nozzle experiments.
Our results
show that the gas density distribution for
OCS in He can include a cone structure superimposed on the near-isotropic background.
Table 4
Tipped Nozzle Doppler Splitting Data for OCS
J = 0 •> 1 Transition at 12163 MHz.
* Angle 8
Splitting (kHz)
50°
70°
0°
37°
Gas carrier
Kr
24.9 + 1
27.7 + 1
Ar
40.0 + 1
42.7 + 1
„
T
3
12
He
90°
O
9 0 + 4
4 2 + 1
4 0 + 1
40+3
90+4
92+3
99+5
+3
102
+8
35
+6
4% OCS mixtures at 2.36 atm. source pressure.
0 + 8
112
splitting values listed at 37° for Ar and He, and at 50° for
He, we refer to Figs. 23 and 24.
In Fig. 23 we show a power
spectrum taken for OCS in He at a nozzle angle of 37°. The
exact 0 -»• 1 frequency, v , is indicated at the mid-point of the
two sets of peaks.
The outer two peaks, separated by 102 kHz,
result from the isotropic, or near isotropic cosp(0-6 ) distribution in the beam, as has already been discussed.
The
inner peaks, separated by 33 kHz, and again symmetric about
v
are only observed in He at this 37° nozzle orientation.
In Fig. 24 we show a power spectrum taken under conditions
identical to those of Fig. 23, except with the nozzle now at
50°.
Evidently the two inner peaks from Fig. 23 have merged.
The two outer peaks remain stationary with a 103 kHz separation.
No sets of inner peaks have ever been observed with
the nozzle at 0°. The obvious explanation for the orientationdependent inner structures, considering the azimuthal symmetry
of the nozzle about its own axis, is a cone with apex halfangle a, chosen such that
2 sin(a-37°)xl3.7 ' 10 cm/sec x 12163 • 106sec
3 . 10
cm/sec
_
35
kHz
(140)
giving a = 55°. At 0 = 50° then, we would expect a splitting
of 10 kHz, which would appear as a single peak at v , as in
Fig. 24.
It is useful to note that, referring to Eq. (136),
81x1-1
TJtH "
50
° '
<141>
to
Figure 23. Power spectrum obtained from OCS J = 0 •*• 1 with 4% OCS
in He at the nozzle angle 9 = 3 7 ° .
o
frequency v
The OCS transition
is located at the exact center of the peaks.
The outer two peaks, separated by 102 kHz, are always
present for OCS in He at any nozzle angle 0 . The inner
set, separated here by 33 kHz, take on this splitting
value only for 9Q = 37°.
FREQUENCY
115
V
0
FREQUENCY
Figure 24. Same gas mixture as that used for spectrum in Fig. 23,
except now the nozzle is tipped at angle 8 = 50°.
The two outer peaks are still present, separated by
102 kHz, but the inner two peaks are merged.
116
so that when present, the cone does not produce additional complication in Doppler pattern when the nozzle is positioned at
0°.
This cone structure is most pronounced in He, although
even in He not present in every gas pulse.
We have observed
weak inner structures in Ar at 9 = 37° as indicated in Table 3,
but these are small in comparison to the outer set.
No such
structures have ever been observed in Kr.
We have modified expression (131) to calculate lineshapes for a thin cone with a = 50° at various nozzle orientations 0 . We show one such time-domain envelope for 0 =
4
0° in Fig. 25. Here v Q = 4 • 10 cm/sec, v = 10 GHz, and
q+1 = 30. The period of 49.6 ysec is nearly identical to
the 48.9 ysec expected for a narrow beam traveling at this
speed at 50° with respect to the vertical axis.
Consideration
of the beam waist terms in Eq. (131) indicates that the cavity
discriminates against signal from the sides of the cone,
effectively producing a geometry not greatly different from
a narrow beam.
Note that L /L, = 0.5 in Fig. 25. We find
that when a = 50° the Doppler splitting scales with 0 Q as
sin(ct-0o)/ as has been assumed in Eq. (140).
In summary,
the experiments listed in Table 3 indicate that the Doppler
splitting is due primarily to an isotropic or near isotropic
background, but that a cone structure with apex half-angle
55° may be formed in He, and to a much lesser extend, in Ar.
This cone structure should not affect the splitting pattern
in the normal 0
= 0° configuration.
Figure 25. Envelope function Eq. (131) calculated for a cone gas
4
distribution with 8^ = 0° and a — 50°, v = 4*10 cm/sec,
o
o
v Q = 10 GHz.
118
We finish by considering a second alternate nozzle configuration, this one with the nozzle placed just behind the
center of one mirror at (0, -d/2, 0 ) , with the beam directed
out along the cavity axis.
In Fig. 26 we show an envelope
calculated for p = 0.5, q+1 = 32, v = 10 GHz, and v = 4*10
cm/sec.
This should be compared to Fig. 12 which was calcu-
lated using these same four parameter values, but with the
nozzle at its normal position.
We find that in going from the
perpendicular to the on-axis position the splitting changes
from 20.5 kHz to 26.7 kHz, which is just equal to 2 v v/c.
The dephasing time is greater in the on-axis case, as is
expected from considering again the effect of the beam waist.
Experimentally, for
84
KrH 35 Cl J = 1 -> 2 at 4802.9 MHz we
find splittings of (10.2 + 2) kHz and (11.7 + 2) kHz for
the perpendicular and on-axis experiments, respectively. Although the on-axis arrangement should provide resolution
superior to the perpendicular case, we have not implemented
it because of practical problems related to the input coupling
of the microwave radiation.
Up to now we have treated the Doppler splitting as a
tool for studying the gas dynamics.
The molecular spectro-
scopist however, is more intent on eliminating the phenomenon
rather than using it for this purpose.
Several suggestions
for this have already been mentioned in the paper.
TEM
In the
cavity mode, with only a half-wavelength between the
mirrors, the splitting will disappear.
Implementing this
to
o
Figure 26. Envelope function Eq. (131) calculated for a gas distribution
(cos0 5 0)/r2 with the nozzle positioned at (0, -d/2,0) and
pointed along the cavity axis. The envelope was calculated
for the same set of parameters as used for Fig. 4.
121
122
method in the Fabry-Perot cavity at microwave frequencies,
where the dispersion relationship is given in Eq. (118),
is obviously impractical.
The key idea here however, is the
half-wavelength, and by choosing a different type of cavity
that could be operated near cutoff, the splitting could be
36
eliminated.
By considering the discussion leading to Eq.
(134) it is clear that the splitting is not restricted to
standing wave structures, and cannot be removed by carrying
out the experiment in a traveling wave structure, as for
example, by using a set of microwave horns.
The idea of restricting the beam so that all molecules
travel perpendicularly to the radiation field has been suggested in several discussions.
We have tried this experiment
by placing a skimmer in front of the nozzle.
Our skimmer was
a metal sheet with a rectangular slot that permitted molecules to travel only in the xz-plane (see Fig. 9 ) . An OCS
J = 0 •*• 1 line obtained with this arrangement is shown in
Fig. 27, taken under conditions nearly identical to those
used for the spectrum in Fig. 12, except that a 20% OCS mixture in Ar was needed just to obtain a reasonably strong
signal.
The severe loss in signal prevented us from ever
using a skimmer while studying new resonances.
Although we have been unable to eliminate the Doppler
splitting in any general manner, the direct proportionality
between Av and v can be exploited by taking rotational spectra
at sufficiently low frequencies that the two Doppler peaks
l->
to
Figure 27. Time domain signal from OCS J = 0 •»• 1 converted at 0.5
ysec/point. The spectrum was obtained with the geometry
of Fig. 1, but with a skimmer in place that nearly eliminated molecular motion along the cavity axis. Very
little doubling structure remains on this line. The
envelope is instrumentally distorted at short times.
124
s
.
o
o
o
00
o
CO
O
w
<0 JJJ
?
a
I
125
merge.
As will be discussed in the
next chapter, this
technique, which has several other important advantages,
has enabled us to assign rotational transitions for approximately ten different molecules whose spectra are complicated
by the presence of two nuclear quadrupole coupling interactions.
In those cases when Doppler doubling is unavoidable,
it is possible, as shown in Figs. 18-21, to select a beam dis
tribution that gives two very sharp, well resolved peaks.
For the reasons discussed above, these peaks can be regarded
as fixed.
126
Chapter IV.
A.
Rare Gas Nuclear Quadrupole Coupling in
van der Waals Molecules
Introduction
Van der Waals molecules are combinations of atoms and
ordinary small molecules held together in part by atom-atom
or atom-molecule attractions that are hundreds of times
weaker than ordinary chemical bonds.37 '38 Some examples are
Ar 2 , ArXe, or any combination of two rare gas atoms, pairs
of ordinary molecules such as (H 2 ) 2 , (N0) 2 , and rare gasmolecules pairs, all of which have well defined electronic,
vibrational, and rotational states that can be characterized
by experimental and theoretical techniques.
of the thermodynamic and transport
Although studies
properties of gases have
for many decades given indirect evidence for the existence of
many van der Waals complexes, it has only been within the last
fifteen years or so that detailed structural and physical information about these complexes have become available.
Most of
this information comes from scattering experiments, infrared
studies, and most importantly, pure rotational, or microwave
spectroscopy.
Experience indicates that it is possible to
form a van der Waals molecule from virtually any two ordinary
molecules simply by mixing the two species together in the
gas phase, along with a rare gas propellant if necessary, and
expanding the mixture through a small orifice into a vacuum.
The measurement of nuclear quadrupole coupling constants
has long been recognized to be a powerful tool for obtaining
127
i n f o r m a t i o n a b o u t t h e d e t a i l e d a r r a n g e m e n t s of e l e c t r o n i c
w a v e f u n c t i o n s n e a r t h e n u c l e i b e i n g s t u d i e d . 39-42 We r e p o r t
83
t h e measurement and a n a l y s i s of t h e
83
quadrupole coupling constants in
1 31
3^
XeD
Kr and
131
Xe n u c l e a r
35
83
KrH Cl, K
D
35
c,
d
Cl. The rare gas nuclear quadrupole coupling is ob-
served through the coupling of the nuclear angular momentum to
the rotational angular momentum of the molecule as measured in
the vibrational ground state, pure rotational transitions of
KrH(D)ci
and XeDCl. The rotational spectrum is observed by
using the newly developed experimental method of pulsed Fourier
transform microwave spectroscopy in a Fabry-Perot cavity with a
pulsed supersonic gas expansion.
Because the nuclear quadrupole
coupling constant of the closed shell free rare gas atom is
identically zero, the effect measured here arises from a distortion of the spherical symmetry of the atom upon complexation.
We thus have a sensitive probe for studying small changes
in the electronic environment of the rare gas nucleus that
occur when the rare gas atom binds to the hydrogen halide molecule. Since our report 23 of the first measurement
of a rare
gas nuclear quadrupole coupling constant in the 83KrHE van der
Waals molecule, we have reported similar measurements in
"^eH^Cl
J
and
°KrDF.
These studies have established,
based on calculated values of the Sternheimer quadrupole
shielding constants 45-47 for Kr and Xe, that most of the
magnitude of the rare gas quadrupole coupling constant in
128
each of these molecules can be attributed to quadrupole shielding effects occurring in the rare gas atom in the presence of
the electron field gradient of the partner hydrogen halide
83
molecule. By combining our experimental results for
KrH(D)F
with the 83KrH(D)35Cl measurements reported here, we establish
this result empirically for Kr, thereby obtaining an experimental estimate for the Sternheimer parameter for this atom.
We also report an improved measurement of the Xe nuclear quad131
3^
rupole coupling constant in
sults for
XeH(D)
XeH
Cl and show that our re-
Cl are consistent with the interpretation
used for the Kr containing species.
B.
Experimental
The experimental technique used here is the method of
pulsed Fourier-transform microwave spectroscopy carried out in
a Fabry-Perot cavity with a pulsed supersonic nozzle gas ex20
pansion.
The source gas for the nozzle consisted of mix-
tures of 0.3% to 1.1% HC1 or DC1 with 70% to 80% He and 20%
to 30% Xe or Kr at pressures ranging from 0.17 to 2.5 atmospheres.
A pulsed solenoid valve with a 0.040 inch diameter
orifice plate was used to pulse the gas into the cavity at
repetition rates of about 1 Hz.
Careful optimization of the
gas mixture and source conditions was crucial for obtaining
adequate signal-to-noise.
When the expanding gas, which is
now at temperatures of 1-10K and contains large numbers of
weekly bound molecular complexes, passes between the FabryPerot mirrors a TT/2 microwave pulse is used to polarize all
129
rotational transitions within the bandwidth of the cavity.
After the polarizing radiation dies away these polarized rotational transitions emit coherently at their resonance frequencies.
This coherence decays because of Doppler-effects,
molecular collisions, and the molecular transit out of the
cavity.
The decaying molecular emission is coupled out of the
cavity and detected in a superheterodyne receiver by mixing
with a local oscillator.
The signal is digitized and stored
to be averaged with the signals from repeating gas pulses.
After an adequate signal-to-noise ratio is obtained the time
domain signal is Fourier transformed into the frequency domain.
In the case of a single resonance frequency, the gas flow from
the nozzle and the resultant Doppler effect gives rise to a
symmetric doublet in the frequency domain.
The gas dynamics
of the pulsed nozzle, the molecular polarization and subsequent emission processes, and the characteristic Doppler
doubling phenomenon have been described in detail in Chapters
II and III.
Most of the data presented here for molecules containing two interacting nuclear quadrupole moments was taken at
48
frequencies below 8 GHz.
Because the Doppler splitting
in the frequency domain is proportional to the resonance
frequency, at these lower frequencies the two Doppler peaks
become merged.
This simplifies the spectrometer signals and
doubles the signal-to-noise ratio.
Furthermore, the quad-
rupole splitting patterns are simplified and better resolved
130
at lower values of the angular momentum quantum number J.
The
deuterium nuclear quadrupole splittings remained unresolved,
causing the resonance lines to broaden.
In Fig. 28 we show a
power spectrum containing the J = 1 -> 2, F^ = 5/2, F 2 = 7,
F = 8 -»• P1* = 7/2, F2* = 8, F' = 9, and F1 = 5/2, F 2 = 6,
F = 5 + F ^ = 7/2, F 2 ' = 7, F* = 8 lines of
83
KrD 3 5 Cl.
The
spectrum was obtained by averaging 30 emission signals,
weighting the resultant time domain record with a digial exponential filter, and taking the power spectrum.
In Fig. 29 we
show a portion of the calculated pattern in the frequency
83
3 *5
domain of the
KrD Cl J = 1 •*• 2 transition. The envelope
83
35
was generated from the measured values of the
Kr and
Cl
nuclear quadrupole coupling constants, and the calculated projection of the deuterium quadrupole coupling constant in free
D 35Cl. The nine resonance lines reported here are represented
as vertical bars.
The assignments are identified in Table 6.
C.
Rotational Spectra and Spectroscopic Constants
for the Rare Gas Hydrogen Halides
83
35
The rotational energy levels for
KrH(D) Cl and
131
35
XeH(D)
Cl are determined by the K=0 symmetric top Hamil-
tonian
E H(J ' XCV XR' V
•*--*• 2
I
F
l ' F 2 'F ) = B o J
3 -»•
" V
,
•*••*• 2-> 2.
2li(2Ii-l)(2J-1)(2J+3)
(142)
'
H
Figure 28. A power spectrum showing the J = 1 •*• 2, F, = 5/2, F 2 = 7,
F = 8 -*- F^
= 7/2, F2' = 8, F' = 9, and F^^ = 5/2, F 2 = 6,
F = 5 -*• F^^' = 7/2, F2' = 7, F' = 8 lines of
83
KrD35Cl at
4.7 GHz. Both the deuterium nuclear quadrupole splittings
and the Doppler splittings are unresolved. This spectrum
was taken in approximately one minute.
T
4752.700
•
n
4752.800
1
r
4752.900
4753.000
(MHz)
to
to
CO
Figure 29. A calculated envelope and the measured transitions for
the 83KrD35Cl J = 1 + 2 multiplet. The assignments are
identified in Table 6.
134
N
X
135
where B~o = 7?
z (B„
o + C O) is the rotational constant, DJ is the
centrifugal distortion constant, and X-j and !••» i = 1, 2, 3,
are the chlorine, rare-gas, and deuterium nuclear quadrupole
coupling constants and nuclear spins, as necessary.
Matrix
elements of this Hamiltonian were calculated to first order in
the basis
* + Icl = F r
F-L + I R = P2/
•*•
•*•
(143)
-*•
F 2 + I D = F,
where the subscript "R" indicates either Kr or Xe.
I i*j1
XeH
Cl data was fit with energy levels calculated from
Eq. (142) only to first order.
Small second-order chlorine
nuclear quadrupole effects in the
and
The
O C
XeD
KrH(D)
Cl and
131
XeD 3 5 Cl
Cl spectra were taken into account by adjusting
the appropriate energy levels obtained from the first-order
evaluation of Eq. (142) by amounts corresponding to the second order corrections obtained by direct diagonalization of
the single chlorine quadrupole Hamiltonian.
Except in the
83
35
case of the F,=3/2->-F1,=3/2 series of transitions in
KrH Cl,
where a second order shift of 9 kHz did effect the fit to
Cl
X / second order shifts were 4 kHz or less.
Calculated line positions for all deuterium containing
molecules were obtained as weighted averages of all unresolved
components estimated to contribute to the corresponding
136
measured frequencies.
Since some of these lines consisted of up
to ten unresolved hyperfine lines, the assignments are in certain cases identified only by the first, last, and largest contributing components.
The observed spectra, their assignments,
calculated frequencies, and frequency differences, are listed
in Tables 5-8.
The spectroscopic constants used to fit the
data are shown in Tables 9 and 10.
of
The spectroscopic constants
KrHC
N, taken from a report on the microwave spectrum
and structure of this molecule,49 are listed in Table 11. The
structural parameters, R , the distance between the rare gas
nucleus and the center-of-mass of the hydrogen halide, and 0,
the rare gas—center of mass of the hydrogen-halide—hydrogen
(deuterium) angle, are listed for KrKCl and XeDCl in Table 12.
These structural parameters are shown in Fig. 30.
The angle
Y between the a-inertial axis and the hydrogen-halide bond is
23
obtained in the usual way
by assuming that the measured
value of xCl is given by the projection of the chlorine nuclear
quadrupole coupling constant x Q
Cl
of
free H
35
Cl onto the a-
inertial axis according to
X C1
.
X Q
C1 ( 3 C O S ^
,
( 1 4 4 )
where the brackets indicate averaging over the ground vibrational state of the complex.
If the hydrogen halide bond
length r remains unchanged upon complexation, then R Q and 0
may be obtained from y and the rotational constant B .
The assumption here regarding r, and the assumption that the
137
Table 5
Observed and Calculated Frequencies for
J+J'
2
2F
1 2
1
10
1
10'
1
10
1
8
I±
2F '
2F'
Observed
(MHz)
83 35
KrH Cl
Calculated
(MHz)
Difference
(kHz)
f 4819.3741
4819.3689
5.2
1
8
1
1
8
1
3
12
5
14
4819.6873
4819.6885
-1.2
5
14
7
16
4819.9086
4819.9061
2.5
5
6
7
• 4819.9687 4819.9723
-3.6
5
12
7
14
5
10
7
12
4820.1545
4820.1524
2.1
5
8
7
10
4820.2222
4820.2217
0.5
3
8
5
• 4820.2581 4820.2459
12.2
5
6
7
8
3
12
3
10
4825.0474
4825.0454
2.0
3
12
3
12
4825.0872
4825.0878
-0.6
3
8
3
10
4825.4199
4825.4184
1.5
3
8
3
6
4825.4681
4825.4714
-3.3
3
10
3
10
4825.5646
4725.5705
-5.9
3
10
3
8
4825.5912
4825.5857
5.5
10
8
-
8^
138
Table 6
Observed and C a l c u l a t e d F r e q u e n c i e s f o r
J->J'
U-7-U
2F 2F 2F 2F ' 2F ' 2F
_1 _ !
—A " 2
1 2 a
o b s e r v e d
(MHz)
83
35
KrD' C l
Calculated
(MHz)
Difference
(kHz)
3
6
4
5
6
4
4752.2616
4752.2591
2.5
b
3
12
14
5
14
16
4752.4514
4752.4565
-5.1
c
5
4
6
7
2
4
4752.6426
4752.6366
6.0
d
5
14
16
7
16
18
4752.7641
4752.7656
-1.5
5
12
10
7
14
12l
5
12
14
7
14
16 ' 4 7 5 2 . 8 4 9 2
4752.8545
-5.3
5
10
10
7
14
12,
3
8
8
5
6
6>
5
10
8
7
10
4753.0087
-2.4
5
12
12
7
10
12.
5
10
12
7
12
U
5
10
8
7
12
10 ' 4 7 5 3 . 1 1 0 6
4753.1061
4.5
5
10
10
7
12
12,
h
5
8
10
7
10
12
4753.1979
4753.2012
-3.3
i
3
10
12
5
12
14
4753.5214
4753.5171
4.3
e
f
g
8 • 4753.0063
139
Table 7
Observed and C a l c u l a t e d Frequencies for 131 XeH35Cl
2F X
2F
2
V
2F'
Observed
(MHz)
Calculated
(MHz)
Difference
(kHz)
5
6
7
8
3963.9586
3963.9587
-0.1
5
4
7
6
3964.0333
3964.0335
-0.2
5
8
7
10
3964.1764
3964.1749
1.5
3
6
5
8
3964.3899
3964.3911
-1.2
5
6
7
8
5944.9615
5944.9625
-1.0
7
6
9
5945.0682
5945.0703
-2.1
5
4
7
7
8
9
5945.1482
-0.7
4
9
"}
5945.1475
7
7
10
9
12
5945.2383
5945.2373
1.0
5
8
7
10
5945.3294
5945.3297
-0.3
: )
140
Table 8
131
35
Observed and Calculated Frequencies for
XeD Cl
J-+J' 2F 1 2F 2 2F 2Fn ' 2F 0 ' 2F' Observed Calculated Difference
(MHz)
1
(MHz)
(kHz)
5
2 3
6
-4.0
8
5856.5740 5856.5780
7
7
6
4
9
7
6
-2.3
8
5856.7137 5856.7160
9
5
4
4
7
7
8 10
9
7
-3.7
8
5856.8119 5856.8156
8
9
7
4
4
9
7 10
-0.5
5856.9279 5856.9284
8
9
5
8
6
7
0.3
5857.0395 5857.0392
5
8 10
7
5
12.8
5857.0665 5857.0537
8
8
7
5
-5.6
5857.1040 5857.1096
2
4
7
5
3 4
8
6
7
5
-0.3
7807.2187 7807.2190
8 10
7
5
2
4
7
7
6
4
9
7
-5.5
7808.2831 7808.2886
8 10
9
7
8
8
9
9
8 10 11
6.5
7808.3202 7808.3137
9
8
8 11
9
6
8 11
-8.2
7808.3538 7808.3620
9
6
6 11
9 10 12 11
0.5
7808.3722 7808.3717
9 10 10 11
-1.2
7808.4308 7808.4320
9 12 10 11
-1.5
7808.4901 7808.4916
7 10 12
9
7
5
6 7 11
9
4 5
10
8
14
12
9
7
13
11
9758.9873
9758.9855
1.8
13660.7125 13660.7133
-0.8
Table 9
Spectroscopic Constants of KrHCl
Isotope
83
KrH 35 Cl
83 K r D 35 c l
B0-8Dj(MHz)
Cl
x
(MHz)
xKr(MHz)
D
X
<MHz>
1204.84742(40) -29.238(45) 5.200(100)
1188.01053(60) -40.824a
7.192(100)
0.1135b
a
83
3K
84
3^
Fixed
at
average
of
the
KrD
Cl
and
KrD
Cl values from
Ref. [23].
b3Fixed at t)
35
Fixed
at
the
projection
of
the
free
D
Cl deuterium quadrupole coupling constant.
Table 10
Spectroscopic Constants of XeHCl
Isotope
B v(MHz)
o ~ "'
DT(kHz)
~J
x
(MHz)
x
(MHz)
132
XeD 3 5 Cl
974.50768(35)
3.4209(20)
-44.800(40)
131
XeH 3 5 Cl
990.86302(32)
3.796(20)
-34.76b
-4.641(50)
131
XeD 3 5 Cl
976.11556(40)
3.4285(67)
-44.780(200)
-5.89(20)
X
(MHz)
0.1245*
0.1245*
a
3s
Fixed at the projection of the free D Cl deuterium quadrupole coupling constant.
b
Fixed at the value for
C1
X
in
129
XeH 3 5 Cl from Ref. [43].
143
T a b l e 11
S p e c t r o s c o p i c C o n s t a n t s of KrHCNa
Isotope
83
a
KrHC l 4 N
Ref.
[49].
B -8DT(MHz)
o
J
1184.60700(20)
XN(MHz)
-3.2630(60)
xKr(MHz)
7.457(50)
Table 12
Structural constants of KrHCl, KrHF,
KrHCN, XeHCl, and XeHF
Isotope
RQ(A)
0 (deg
83
KrH 35 Cl
4. 0824
38.07
83
KrD 35 Cl
4. 0652
31.03
83KrHF
3.6076
39.17
83KrDF
3. 5575
31.27
83
4. 5203
27.50
KrHC 14 N
131
XeH 3 5 Cl
4.2456
34.76
131
xeD 3 5 Cl
4. 2259
28.37
131
XeHF a
3..7772
35.7
131
XeDF a
3..7339
29.55
a
Ref. [52].
H
center-of-mass
a-axis
Cl
A = Kr, Xe
Figure 30. Structural parameters for the rare-gas hydrogen
chloride series of van der Waals molecules.
146
change in x
Cl
from x 0
Cl
is purely a geometrical effect and does
not involve any changes in the electronic environment of the
50 51
chlorine nucleus can be tested experimentally, '
and are
found to be correct for the rare gas hydrogen halides within
experimental error.
. .
The structural analysis for KrHCN is
14
similar, using the projection of the
coupling constant.
N nuclear quadrupole
The structural parameters for this mole-
cule are shown in Fig. 31.
D.
Analysis
The rare gas nuclear quadrupole coupling constant, X/ is
determined by the product of the quadrupole moment Q of the
rare gas nucleus and the electric field gradient, q, at the
nuclear site along the molecular a-inertial axis due to all
charges outside the nucleus,
x isigiven by
where e is the proton charge and h is Planck's constant. Since
the quadrupole moments of the 83Kr and 131Xe nuclei have been
determined to approximately 15% accuracy by atomic hyperfine
structure measurements, the problem of evaluating x reduces
to determining q.
Direct calculations of this quantity from
first principles are difficult, particularly in molecules with
such small binding energies, and do not give a very physical
picture of the origin of the results.
We will follow here
3 9 '40 of trying to
the more useful and conventional approach 1 '
understand the origin of the field gradient by using a few
H
a-axis
oKr
Figure 31.
9 A center-of-mass
Structural parameters for KrHCN.
0 and y differ by 0.6 degrees.
The angles
148
simple parameters characterizing the structural and electronic
properties of the Kr and hydrogen halide subunits, and the
structure of the van der Walls complex.
The sources of possible contributions to q may be divided
into the following categories:
1.
Valence electrons of the rare gas atom,
2.
Distortion of the closed shells of the rare gas atom
by the hydrogen halide partner,
3.
Charge distributions lying outside the rare gas atom.
For atoms participating in ordinary chemical bonds, for example
the halogens Cl, Br, and I in ordinary molecules, contributions
from electrons in the uncompleted valence shell account for most
of the observed field gradients.
Because this contribution is
identically zero in a free rare gas atom, we can expect it to
be small in the very weakly bound van der Waals molecules,
so that the ordinarily negligible effects of the second and
third type can become important. Foley, Sternheimer, and
Tycko have shown47 that the field gradient at the nuclear
site in a closed shell system resulting from an external charge
2e
e at a distance R from the nucleus can be written — j (!"¥„,) i
R
where R must exceed the radius of the atom or ion.
The pro-
portionality factor Yoo is specific to each atom or ion, and
results from the interaction of the electrons in the closed
shells with the perturbing charge.
Since this is a first-
order effect, this same direct proportionality will hold for
an electric field gradient arising from any system of external
charges.
149
83
23.44
131 35
43
In view of our results for ^KrHfDjF^'** and
XeH JD Cl,
we will begin by analyzing contributions to field gradients
at the rare gas nuclear sites due to the direct and shielding
enhanced field gradients arising from the first few electric
multipole moments of the hydrogen halide bonding partners.
The electric field gradient, q , at a point (R,0) along the
radial direction outside a neutral cylindrically symmetric
molecule is given by
/cos9\
yP2(cos8)v
,P3(cos0)i
qo = ^ 2 2 J i ) - 12Q ( - ^ 5 — ) - 20^-1-^—)
(146)
,P 4 (COS0) y
- 30$/
?
) - ...
where u, Q, J3, and $ are the electric dipole, quadrupole, etc.
moments of the molecule.
The brackets here indicate the ex-
pectation values in the vibrational ground state.
All coordi-
nates and moments are referred to the molecular center of mass.
Provided that the electronic wavefunctions of the rare gas
atom and hydrogen halide do not overlap, the field gradient
at the rare gas nuclear site can be written
q - q0 ( 1 - Y J •
(147)
The contribution to x i s then given by
X
=
~
eq n Q(l-Yj
w
°
(148)
The series for q in Eq. (146) was calculated out to
the third term for KrHCl, KrHF, and KrHCN using the values
of R and 0 listed for each molecule in Table 12, and the
150
multipole moments for H
35
14
Cl, HC N, and HF listed in Table 13,
14 and 15. The calculated values of -eqQ/h and the measured
P
X values are listed in Table 16. In Fig. 31 we have plotted
Kr
X
as a function of -eq /h. We find a direct proportionality
Kr
"*
Kr
between x
and -eq /h, with a predicted value of x
of
near
zero in the absence of an external field gradient.
The most immediate problem with this analysis is the
question of the convergence of the multipole expansion.
In
these molecules the center-of-mass separations across the van
der Waals bonds are sufficiently large that the respective
electronic wavefunctions barely overlap. In 83KrD 35Cl for
o
example, the R
of 4.06A compares to the van der Waals radii
o
o
of 2.0 A and 1.80 A for Kr and Cl, respectively. That part
of the perturbing Hamiltonian giving rise to the effect
parameterized by Eq. (147) is given by 53
2
H
pert
=
e(
- R->
V
cos9
a>
10OT (cos0 K )
3ucos0,
^ — ^
+
6QP2(cos0j;))
— ^ T
(149)
15$P,(cos0 K )
4
R*
5
R°
•*•.../
for an e l e c t r o n i n a r a r e gas sheet a t c o o r d i n a t e s ( r , 0 ,
a
ij) ) (See Fig. 33) in the presence of a cylindrically symmetric
charge distribution at a center-of-mass distance R, and
o
orientation 0fa.
Taking R = 4.1 A, 0. = 31°, and using the
35
multipole moments for D Cl, we obtain from Eq. (149), in
relative importance
151
Table 13
Molecular Properties of H 35 C1 and D 35 C1
H 35 C1
D
X
(MH Z )
a
D 35 C1
0.18736
li(D) a
1.1085
1.1033
Q(DA)
3.74 (12) b
3.74d
£2(DA2)
2.446°
2.446d
$(DA3)
4.704°
4.704d
a
E. W. Kaiser, J. Chem. Phys. 53_, 1686 (1970).
F. H. deLeeuw and A. Dymanus, J. Mol. Spect. 48, 427 (1973)
C
D. Maillard and B. Silvi, Mol. Phys. 40, 933 (1980).
d
35
Fixed at corresponding H Cl value.
152
Table 14
Molecular Properties of HC
u(D) a
2.9846
Q(DA) b
2.42(60)
fi(DA2)C
4>(DA3)°
a
14
N
"
6
'366
6.422
A. Maki, J. Phys. Chem. Ref. Data 3_, 231 (1974) .
b
°
Average of calculated values of 2.12 DA, J. Tyrrell, J.
Phys. Chem. 83, 2907 (1979), 2.03 D&, ref. [54], and an
experimental value of 3.1(6) DA" measured for Hcl5N by S,
L. Hartford, W. C. Allen, C. L. Norris, E. F. Pearson,
and W. H. Flygare, Chem. Phys. Lett. 18, 153 (1968).
c
Ref. [54] .
153
Table 15
Molecular Properties of HF and DF
a
HF
DF
y(D) a
1.8265
1.8188
Q(DA) a
2.36(3)
2.32
n(DA2)
1.699b
1.699°
°3
$(DA°)
1.804b
1.804°
See Ref. [44].
*u
D. Maillard and B. Silvi, Mol. Phys. 4_0, 933 (1980).
°Values for DF assumed to be identical to those for HF.
Table 16
Summary of Measured and Calculated Quantitites
Related to the Rare Gas Coupling Constants in
KrHF, KrHCl, KrHCN, XeHCl, and XeHF
Isotope
XR(MHz)
eq(calc'd)/h(MHz/b)
83
KrH 35 Cl
5.20(10)
0.2629(402)
83
KrD 35 Cl
7.19(10)
0.3494(570)
83KrHF
10.23(8)
0.4987(575)
83KrDF
13.83(13)
0.6706(880)
7.46(5)
0.3971(1013)
83
KrHC 14 N
131
XeH 3 5 Cl
-4.64(5)
0.2522(388)
131
XeD 3 5 Cl
-5.89(20)
0.3166(504)
131XeHF
-8.59 a
0.4578(570)
131XeDF
-10.57 a
0.5616(710)
a
Ref. [52].
Ol
Figure 32. The measured
Kr nuclear quadrupole coupling constant
values plotted as a function of the electric field gradient at the Kr nuclear site calculated from the first
few electric multipole moments of the hydrogen halide
bonding partner. The slope of this line is related to
the product of the quadrupole shielding of the rare gas
nucleus, and the nuclear quadrupole moment, and the intercept can be related to the amount of charge transfer
occurring in these systems.
16
14
KrDF
12
10
83
8
X
X
KrHC,4N
83
6
KrD35C.
83
KrH35CI
4
0
-2
0.0
1
0.1
0.2
0.3
0.4
- e q (calc'cD/h
1
0.5
0.6
(MHz/b)
0.7
0.8
•A
e
center-of-mass
-H
R
Figure 33. The coordinate system used to parameterize the multipole
expansion of the energy of an electron in a rare gas shell
at coordinate (r, 0 , 4> ) in the field of a cylindrically
cl
cl
symmetric system of charge at a center-of-mass distance R.
j- 1
tn
158
H
pert
=
°* 86
+ 1
'°
+
°* 13 -
No experimental results are available for /p.(cos0)\ for
i >^ 3, and algebraic calculation of these terms using arc
9
I/a
cos ( (cos 0)0/
is not useful for i > 4.
The behavior of these
higher order expectation values can probably be estimated using
hindered rotor wavefunctions 55 for the hydrogen halide portion
of the molecule. When the dominant term in the angular potential in the region being sampled by the hydrogen atom is of
the form l-cos0, the expectation values for P.(cos8) decrease
monotonically to zero as i increases.
using a potential for the form 113 cm
For example, in
KrD
Cl,
(l-cos0) with hindered
35
rotor wavefunctions to describe the D Cl subunit, those authors
calculate
P-(cos0)
= 0.85,
P,(cos0)
1
= 0.61,
P,(cos0)
= 0.37,
P4(cos0)
= 0.19.
(151)
The experimental results for the first two expectation values
P, and P 2 are 0.88
and 0.61.
Algebraic expressions for
P, through P 4 yield 0.86, 0.60, 0.29, and -0.02, respectively
for 0 = 31.0°.
The behavior of the <P (cos9)/ values as
suggested by Eq. (151), which contrasts sharply with the
behavior of algebraically evaluated values of P.(cos6), i >_ 4,
which oscillate between +1 and -1, is likely to make an
important contribution to the very rapid convergence of the
159
series in Eqns. (146) and (149) . Using the value from (151)
for ^P^tcosO)^ to calculate the next term of the series in
Eq. (150) we obtain 0.012.
Other uncertainties in applying
Eq. (146) can be estimated more directly, and these are indicated as error bars in Fig. 32.
Because the a-inertial axis
in these molecules is nearly aligned with the center-of-mass
axis, errors in ^P2(cos0)S will be negligible.
An experimental
value for ^P,(cos0)/ can be obtained from electric dipole measurements a s 5 6 ' 5 7
M(complex) = u(diatom){P±(cos0)) (1 + 2a,
=|) ,
R
o
(152)
where the hydrogen halide dipole moment is projected onto the
a-axis, with a correction for the polarizability, a, of the
rare gas atom.
Because (/cos Y ) )
from Eq. (144) and (COSY/
as obtained from Eq. (151) differ by no more than 2-3% in
KrH(D)Cl,
and XeHCl,
we have chosen to calculate (cos0/
algebraically from ( P 2 ( C O S Y ) } .
Except in the case of KrHCN,
errors in the hydrogen halide molecular quadrupole moment
contribute uncertainties in q of 1-2%.
Errors arising from
the neglect of terms higher than PgtcosO) when calculating
Eq. (146) have been estimated as
2OJ2<P2(cos0))
R6
o
(153)
With these qualifications considered we take up the
physical interpretation of the curve in Fig. 32. Within the
160
uncertainties assigned to the calculated values of -eq /h,
the effect is first order, as is expected from the estimated
importance of dipole polarization 43 '47 and second-order
quadrupole shielding effects.47
A linear least squares fit
to the data in Fig. 32 yields a slope of (21.2+4) barns, and
a frequency axis intercept of (0.376+1.6) MHz.
The estimated
uncertainties in the slope and intercept include the uncertainties in the values of -eq /h.
The direct proportionality observed here between q and
q
can readily be understood qualitatively as arising from
Sternheimer-type quadrupolar shielding occurring in the Kr
atom.
We emphasize however, that we do not know enough about
the very short parts of the Kr and hydrogen halide interaction
to be able to reliably convert this constant into Ye f° r K r *
It is well known47 that in the idealized case of a closed
shell system perturbed by a point charge at distance R, the
shielding parameter has a functional form Y ( R ) / with limiting
Y^* F o r moderately heavy
+
+
—
47
systems such as Cs , Rb , and Cl , Sternheimer et al. found
that 1~Y(R) attains approximately 70-80% of its asymptotic
values of y(Rr=Q)-0,
and
Y^"* 00 )
=
o
value of 1~Y
for R - 2 A, with most of the contribution at
o
R > 2A coming from the valence p shell of the ion.
Although
our proportionality constant is empirically well established,
it is probably related in a complicated way to the wavefunctions of the hydrogen halide, and to the functional form Y ( R )
for the rare gas atom.
It is still interesting to convert our slope into an
effective shielding constant for Kr in the Kr-hydrogen halide
systems studied here.
Using Eq. (148) and the nuclear quad59 60
83
rupole moment of 0.27b
for the
Kr nucleus, we obtain
an estimate of (-77.5+15) for the Kr shielding parameter,
t
subject to the serious qualifications outlined above, and not
83
including uncertainties of perhaps 15% or less in the
Kr
Estimates of Yoo f o r
44
Kr include a non-relativistic variational calculation
including exchange effects for radial perturbations,
giving ym =
62
nuclear quadrupole moment measurement.
-68, a non-relativistic frozen-core calculation
giving Yoo =
-67, estimated by those authors to be in error by perhaps
15%, not including errors arising from neglect of relativistic
fi 3
effects, and a calculation
based on relativistic HartreeFock-Slater electron theory, giving Yro = -84. Relativistic
64
effects have been estimated elsewhere
to account for approximately 7% of this last result.
The vertical axis intercept of the line in Fig. 32
(0.376+1.6) MHz indicates that the combined effects of orbi65
tal overlap
and charge transfer from the Kr valence p
orbitals is small.
We may obtain an estimated upper limit
to the amount of charge transferred from the Kr 4p orbital
aligned with the molecular axis by noting that the transfer
of one electron out of this orbital would result in an
electric field gradient at the Kr nuclear site along the
44
a-inertial axis of magnitude 750 MHz.
The 1.6 MHz error
162
bound on the intercept then corresponds to a fractional elec_3
tron transfer of 3*10 , or less. This result is consistent
with our previous estimation 43 '44 that charge transfer makes
a negligible contribution to the quadrupole coupling constant
in KrH(D)F.
In Fig. 34 we have repeated the analysis used for the
krypton data. We have included here measurement of the 131Xe
nuclear quadrupole coupling constant reported by Baiocchi et
al.
for
Table 16.
XeHF and
XeDF, which are also listed in
Fitting a line to the data shown in Fig. 34 we
obtain a slope of 19.17 + 4 barns, and a frequency axis
intercept of -0.183 + 1 . 4 MHz. Using the 131Xe nuclear quad6 6fi7
rupole moment of -0.12b,
'
we obtain an effective Xe
shielding parameter of -158.8 + 33.
for Xe are -138, 4 3 -130,
and -177,
Calculated value of Y ^
with relativistic
effects accounting for about 19% of this last value. Using
a conversion of 374 MHz/electron appropriate for Xe,43 we
obtain an estimated upper bound of 0.004 electrons transferred
from the Xe atom to the hydrogen halide, consistent with our
measurement for the Kr system.
The validity of these conclusions regarding change
transfer rely heavily on the empirically observed linear
relationship between the measured coupling constants and
the multiple expression for the electric field gradient.
Although the reasons for the apparent usefulness of a long
range expansion at these very short distances between the
rare gas and hydrogen halide molecules have never been
Figure 34. The measured
Xe nuclear quadrupole coupling constant
values plotted as a function of the electric field
gradient at the Xe nuclear site calculated from the first
few electric multipole moments of the hydrogen chloride
bonding partner. The XeHF and XeDF values are taken from
Ref. [52] .
164
i
T
mm
1
' "|
1
^w
_
O
m
O
—
a0>
-
<P
—
-
.O
•**<
—
x
* •
o
X
•**^
o
o
^-**
OJm
o
\
>
X
—
ro
|
I
i
5
«*—"
ro v.
-o
d o
«° K
3*P\
g
X
£
X
JO
CvJ
*—.
GO
i
(0
I
I
( z HIAI) 9 x X
CVJ
I
OJ
O"
1
165
investigated, here or elsewhere, it is known that similar long
range expressions are valid in two other cases.
The values of
electric dipole moments in these hydrogen-halide rare gas
can be explained in a consistent number by projecting the
dipole moment of the free hydrogen halide onto the a-inertial
19 68
axis of the complex, '
and including terms to account for
the dipole and quadrupole induced electric dipole moment of
the rare gas atom.
Electric dipole moments calculated this
way appear to be consistent with projection angles obtained
from measurements of the halogen coupling constant Eq. (144).
More evidence for the validity of applying ]ong-range
expansions to calculate the properties of these systems
comes from the calculations reported by Keenan et al.
on the bending modes of the hydrogen halides.
'
Expressions
giving the attractive interaction between a polarizable sphere
and a polarizable linear molecule are shown to give accurate
values for the experimentally measured values of ^?2(cos0)}
in these systems.
E.
83
Measurement and Analysis of the
Kr Nuclear
11
Quadrupole Coupling in
3c
go
Kr
C1F
The change transfer limits of less than 0.002 to 0.004
electrons reported here, and the long range electron dipole
and bending mode analyses mentioned above have a direct bearing on a fundamental question underlying the studies of van
der Walls molecules, namely "What determines the structure of
a given van der Waals molecule?"
Can the structures of these
166
systems be understood entirely in terms of what are normally
regarded as long range interactions - electrostatic, induction, dispersion and exchange repulsion forces - or do some
of the concepts applicable to more strongly bound molecules
still remain important even in these very weakly bound systems?
The strongest evidence suggesting that ordinary chemical interaction may be important in rare gas containing van der Waals
systems come from the apparent tendency of the rare gas atom
to act as a weak Lewis base when it interacts with a strong
Lewis-acid.
The most clear cut examples of this tendency
69
70
are the complexes between Ar and BFo
and SO , in which the
Ar binds to the B or S atom, giving C3v- symmetry. This
71
interpretation of van der Waals bonding has been proposed
to provide a qualitative explanation for a large number of
van der Waals complexes, although it is certain that in many
of these cases the detailed calculations and measurements
needed to establish the validity of the polarization
explanations do not yet exist.
In an attempt to exploit the empirical relationships
83
obtained in our study of the
Kr nuclear quadrupole interaction in the rare gas-hydrogen halide system to a molecule
83
not a member of this class, we have measured the
Kr
83 3 ^
quadrupole coupling constant in
Kr ClF. Assignments of
82 35
the microwave and radiofrequency spectra of the
Kr ClF,
84 T%
8fi "^R
84 37
Kr ClF,
Kr ClF, and
Kr 'ClF isotopic forms of this
molecule have been reported by Klemperer and coworkers. 72
167
83
Kr containing isotopic forms of this mole72
cule were reported in that earlier work.
The present study
No results on any
is particularly interesting because although ArUlF and KrClF are
structurally very similar to the rare gas hydrogen halides
(see Fig. 37), with the equilibrium position of the diatom at
0 = 0, and with the more electropositive Cl atom located at the
bonding site, the force constant, k b , for the bending modes in
both cases
are
from 5 to 7 times the force constants in the
rare gas hydrogen halide system (see Table 17).
Considered
in terms of its polarizabilities and electric multipole momentum however, ClF is quite similar to the hydrogen halides.
This apparent discrepancy has prompted suggestions 71 that
this difference in bending behavior results from the substitution of a highly directional, primarily Cl 3p z , accepting
orbital m
ClF for the more isotropic, hydrogen ls-like
accepting orbital in the hydrogen halides.
Although our
quadrupole coupling measurement does not rule out the possibility of a electron donor-acceptor interaction, it does
show that any charge transfer in KrClF is probably below the
level of a few thousandths of an electron, at least a factor
of ten smaller than ordinary chemical amounts.
83 35
The
Kr ClF spectra reported here were obtained with
a 3.4% mixture of ClF in Kr expanded through a 0.040 inch
diameter orifice.
We use a Model 8-14-900 pulsed solenoid
valve manufactured by General Valve Corp.
Careful optimiza-
tion of the gas mixture, source pressure, and orifice size
Table 17
Comparison of Bending Force Constants for Several
Rare-Gas Hydrogen Halide Molecules and Ar and KrClF
kb
(mdyne A)
KrHF
0.0039
ArHCl
0.0015
KrHCl
0.0022
ArClF
0.022
KrClF
0.031
169
were crucial to obtaining adequate signal-to-noise.
By working
at 5.7 GHz we eliminated the Doppler splitting effect and
obtained simplified quadrupole splitting patterns.
In Figure
35, we show a power spectrum of the J = 2 ->• 3 , F, = 7/2 •*• F = 8 •+ F. =
9/2, F' = 9, and F± = 7/2, 7/2, F = 7, 2 •+ F.^ = 9/2, 9/2, F' =
8, 2 resonances at 5576.8 MHz.
This spectrum is the transformed
average of approximately twenty time domain records taken in
about one minute.
The rotational energy levels for
83 35
Kr ClF are determined
by the K = 0 symmetric top Hamiltonian
1 ,-> ->-
E
H(J I
-»•-)•-»•
I
P
' Cl' Kr l'
_ -*2
P)
°V
-*4
D
" JJ
2
yCl[3(fcl.3) +|(fcl*^)
- ^
2
]
(154)
2IC1(2IC1-1)(2J-1)(2J+3)
+ | d K r - J ) - lgrJ2]
2IRr(2IKr-l)(2J-1)(2J+3)
y Kr[3(I K r .J)
2
,
where B
= 2"(B +C ) is the rotational constant, D is the disci
Kr
tortion constant, and x / X / an(^ I ci' '''Kr a r e t n e chlor:i-ne
and krypton nuclear quadrupole coupling constants and nuclear
spins.
Matrix elements of this Hamiltonian were calculated
to first order in the basis
->•
->-
->•
J + I cl = FV
(155)
•+
F
l
->+ X
-+•
F
Kr - '
Second order chlorine nuclear quadropole effects were taken into
account by adjusting the appropriate energy levels obtained
170
from the first-order evaluation of Eq. (153) by amounts corresponding to the second order corrections obtained by direct
diagonalization of the single chlorine quadrupole Hamiltonian.
The observed spectra, their assignments, calculated frequencies,
frequency differences, and second-order corrections are listed
in Table 18. The spectroscopic constants used to fit the data
are listed in Table 19. Also listed here is the value for B
obtained from B
- 18 D_ by assuming a D_ of 1.86 kHz as
determined earlier. 72 In Figure 36 we show a portion of the
calculated pattern in the frequency domain of the 83Kr 35ClF
Cl
Kr
—
J = 2 -> 3 transition using the values of x / X / an<3 B 18 D_.
Measured resonance lines are represented as vertical
bars.
The structure and force field of KrClF are discussed in
detail in reference [72]. In Table 20 we summarize the struc83
tural information needed to analyze the
Kr nuclear quadrapole coupling constant. The vibrationally averaged structure
of KrClF in the vibrational ground state is nearly linear, as
shown in Figure 37, with the chlorine atom lying between the
other two atoms. The angle Y between the a-inertial axis and
Cl
—1
2 1/2
the ClF axis is obtained from
the usual way.
B~0 and Y
are
x
as cos
((cos Y )
)
in
then used to obtain the center-
of-mass separation R and the structural angle 0.
The Cl-F
distance is assumed to be unchanged upon complexation from
its value in free ClF.
Table 18
Observed and Calculated Frequencies for
83 35
Kr ClF
2F -> 2FJ
2F'
Observed
(MHz)
Calculated
(MHz)
Difference
(kHz)
3
12
5
14
5567.6165
5567.6133
3.2
-94.9
1
10
3
12
5567.7178
5567.7197
1.9
48.2
5
14
7
16
5576.5876
5576.5874
0.2
5.5
7
16
9
18
5576.8415
5576.8457
4.2
21.9
7
14
9
16
5576.8721
5576.8681
4.0
21.9
7
4
9
4
7
12
9
12
7
12
9
14
5577.0151
5577.0103
4.8
18.9
5
8
7
8
5
12
7
14
5577.0715
5577.0776
-6.1
5.5
2F 2
}
Second-order
correction (kHz)
to
Figure 35. A power spectrum showing the J = 2 •*- 3, F. = 7/2, F = 8
-»• F^
= 9/2, F2" = 9 , and Fx = 7/2, 7/2, F = 7, 2 •»• F^
= 9/2,
9/2, F* = 8, 2 lines. The envelope is the transformed
average of approximately twenty time domain records. Resolution in 3.906 kHz per point. The vertical bars represent
the appropriate frequency measurements taken from Table 18.
b
5576.800
5576.900
5577.000
(MHz)
^1
Table 19
Spectroscopic Constants for
B Q - 18Dj (MHz)
x C 1 (MHz)
929.2174(10)
-141.519a
a
b
x K r (MHz)
13.90(25)
Fixed at the average of values for
and
86
83 35
Kr ClF
82
Kr 35 ClF,
B~o (MHz)
929.2509b
84
Kr 3 5 ClF,
Kr 3 5 ClF from Ref. [72].
Obtained from B" - 18D_ using D_ = 1.86 kHz from Ref. [72].
o
J
J
tn
Figure 36. A calculated envelope and five of the seven assigned
transition frequencies for the J = 2 -»- 3 transition in
83
KT35C1F.
5576.500
5576.700
5576.900
(MHz)
5577.300
177
Table 20
Structural Constants of
83 35
Kr ClF
o
R cm (A)
3.95525
0 (deg.)
8.62
Y (deg.)
8.11
a-axis
center of mass
R
Figure 37.
Structural parameters for
83 35
Kr ClF.
The
angle y has been shown in Ref. [72] to be
acute.
00
179
The measured rare gas nuclear quadrupole coupling constant, x / i s related to the electric field gradient, q, along
the inertial a-axis at the rare gas site by
XR = - ^
,
(156)
where e is the proton charge, Q is the electric quadrupole
moment of the rare gas nucleus, and h is Planck's constant.
Kr
83 35
The present measurement of x
in
Kr ClF•follows previous
determinations of x K r in
83
KrHC 1 4 N, 4 9
83
83
KrH(D)F, 25 ' 44
KrHC 1 5 N. 4 9
83
KrH(D) 35 C1, 73
We have also measured the
nuclear quadrupole coupling constants in
XeH(D)
131
Cl.
Xe
'
These previous measurements, all involving a rare gas atom
bound to a hydrogen halide, have established that q, calculi
lated from the known values of x
a
nd Q, is directly propor-
tional to the electric field gradient at the rare gas site
calculated from the electric multipole moments of the hydrogen
halide bonding partner and the geometry of the van der Waals
molecule.
We may write
q = qoK,
(157)
where K for Kr has been determined to be 78.5(15) 73 using
—24
2
83
0.27*10
cm for the
Kr nuclear quadrupole moment. Here
q is given by
/cose\
^.P,,(cos8)
(eose).i
•^js—) "
R
- 30 .P4(cos0) »
R
R
^/P,(cos8)
(cosen
20n
^
R
(158)
180
where u, Q, etc. are the electric dipole, quadrupole, and higher
moments of the hydrogen halide change distribution.
All coor-
dinates, moments, and distances are referred to the symmetry
axis and center-of-mass of the hydrogen halide.
The series in
Eq. (158) appears to be converged by the octupole term for all
eight molecules listed above.
The non-unity value of the pro-
portionality constant K may be understood qualitatively as
resulting from Sternheimer type shielding [47] of the rare gas
nucleus by the atomic electrons.
Comparison of our value for
K to calculated values of the Sternheimer shielding parameter
for Kr indicate that the effect seen here is of the correct
magnitude. 43 ' 44 ' 73
The series in Eq. (158) was calculated out to the third
35
term using the values of ]i, Q, and Q, listed for
ClF in
Table 21. The values of the hexadecapole and higher moments
of this molecule are not known.
We obtain contributions to
q of -2.15, -1.84, and -2.80, in units of 1012 statcoulomb
_3
cm , from the first three terms in Eq. (158). Using Eqns.
(156) and (157) with K = 78.5, the estimated contribution
Kr
to x
is 10.83 MHz. The 3.1 MHz difference between this
value and the measured value of 13.9 MHz could be due to
neglected terms in the clearly non-convergent series in
Eq. (158), to Lewis acid-base type charge transfer from the
Kr atom, or to electronic orbital overlap effects.
Consid-
ering the first possibility, if the discrepancy in x results
from a neglected contribution of q of -2.0 • 1012 sc cm-3,
181
Table 21
Properties of 35ClF
a
r(Cl - F) (A) a
1.63178
u(D) b
0.8881
Q(DA) C
1.54(7)
fi(DA2)d
5.83
See Ref. [74].
R. Davis and J. Muenter, J. Chem. Phys. 57
(1972) 2836.
C
B . Fabricant and J. Muenter, J. Chem. Phys.
66 (1977) 5274.
S. Green, a private communication quoted in
Ref. [74].
182
then at least two additional terms in the series in Eq. (158)
would be needed to establish a converging trend in q and
bring it into agreement with the model used to explain the
krypton-hydrogen halide systems. The largest discrepancy in
12
—3
83
14 73
q Q in those systems was 0.6 • 10
sc cm
for
KrHC N.
Because the higher moments of the ClF molecule are increasingly
dependent on the outer portions of the ClF wavefunction, the
validity of using free ClF hexadecapole and higher moments in
Eq. (158) becomes questionable.
We have no way of making accurate estimates of overlap
effects for this complex.
Dipole polarization of the Kr atom
by the electric field of the ClF molecule is a second order
effect and can be estimated to be of negligible importance.43
Experimentally, we find no evidence that electric field
induced field gradients or second order quadrupolar shielding
effects have any importance in the Kr-hydrogen halide systems.
Kr
If the 3.1 MHz difference in x
values is attributed entirely
to charge transfer, we obtain an approximate upper bound of
0.004 for the fractional vacancy in the Kr 4p orbital aligned
39
along the molecular axis, using the Townes-Dailey theory
with a conversion factor of 750 MHz per electron.44 If this
transferred charge were to be located at the fluorine nuclear
74
site, as has been suggested
by analogy to the IC1 2
system,
it would contribute to q by
q ~ -2 x 0.004 e R~3 (78.5),
or about 3-1012 sc cm-3 , a negligible amount. The limit of
0.004 electron is less than a factor of two greater than the
73
183
more reliable upper bound of > 0.003 electron established for
44 73
the krypton-hydrogen halide systems. '
The present study of the 83Kr nuclear quadrupole coupling
constant in KrClF was motivated in large part by the unusual
properties of this complex in comparison to the hydrogen halide
series of molecules, as determined from the study of the normal
isotopic forms of KrClF by Klemperer and co-workers. These
include bonding and stretching force constants 72 that exceed
the corresponding force constants in the Kr-HX series by factors of approximately 10 and 2, respectively, and a van der
Waals well depth that may be as large as 930 cm-1 75 estimated
from a Born-Oppenheimer angular-radial separation analysis
76
of
the vibrational ground state rotational spectroscopic constant.
24
Comparable estimates
of well depths of KrHF, KrHCl, and
—1
—1
83
KrHBr range from 150 cm
to 250 cm
. Our studies of
Kr
nuclear quadrupole coupling constants in five krypton-hydrogen
halide molecules have established a baseline for specific
comparison to
3
KrClF.
In
83
Kr 35 ClF we find that the first
three terms of the electric multipole expansion for the field
gradient at the Kr nuclear site, combined with a quadrupolar
shielding factor for Kr of 78.5, determined by these earlier
studies, can account for 75% of the observed 83Kr coupling
constant m
this molecule.
This gives an estimate of 0.004
for the maximum fractional transfer of an electron from the
Kr 4p
orbital.
This series is not converged at the third
term, and neglected higher terms could easily account for the
184
remaining 25%.
This result appears to be consistent with the
72
electric dipole measurement reported earlier
for KrClF.
Using polarization of the Kr atom by the first three nonzero moments of ClF and back polarization of the ClF by Kr,
72
those authors found
that the calculated KrClF dipole moment
falls short of the measured value by 0.07 D.
Again, the series
does not converge, and they were unable to distinguish between
polarization effects and charge transfer.
The transfer of
0.003 electrons from the Kr atom to F would suffice to make
up the dipole moment difference.
It appears that computa-
tional techniques may have to be used to obtain transfer estimates for KrClF more precise than those reported here.
185
Appendix A
Here e is a well-defined function of three spatial and
one time coordinate (x., x„, x_, x . ) .
e(r - v(r)(t-t'),t')
is formed by taking the composition of e with the functions
x 1 = x - vx(t-t')
x 22 = y - vy (t-f)
(Al)
x3 = z - vz(t-f)
x 4 = t»,
where v , v , and v may themselves be functions of x, y, and
z. We write Eq. (49) as
KfcAn
2-2. sine,
p. (r, v, t) =
where 9 = K/fc
1
at
e (r - v(r) (t-f) ,t ')dt'.
K 2 *An
— 2
3p.
-.
(A2)
Then
cos6{e(r, t)
(A3)
,t ,3e „
~ f. ( 3x7 V x
o
+
K2*iAn
4
3p ±
3x
+
, 9e V „ + , 3eV „+ , 3e
-i-.i
3x7 y
3x7 z
3x7 ' °> d t } '
COse
t
a
£ ^x,
t
o
9v
9x, 3x
a
l
X
(A4)
|e_^+|§_Z)x ( t - t') }df ,
3x 2 3x
3x 3 3x
'
3p.
3p.
with similar expressions for ^-— , -g— . Taking (,-3g-r- + v • V)p.
K AAn Q
yields - — j
cos6e(r, t)
186
2
< AAn
t
- — 4 - ° - cos0 / {
a_
a_
giS. v • Vv x + §§- V * Vvy
+
(A5)
|§- v • Vv
x (t - f )}df ,
2= - ^
e(r, t) An(r, v, t) .
(A6)
The results in Eqns. (33) and (34) with the functional form
v(r) replacing the independent variable v, and in Eqns. (50)
and (59)-(61) may be demonstrated in the same way.
The physical significance of the criterion in Eq. (31)
may be determined as follows.
A particle at coordinate r with
velocity v(r) at time t will at later time t + 6t have moved
to r + v(r)6t.
Then
v v (r + v(r)6t) = v ( r ) + v(r)St ' Vv v
+ .... (higher order terms) = v x (r).
Since the term v(r) • Vv„ in this expansion is zero everywhere
in the cavity, all particles must be traveling on straightline, constant speed trajectories.
187
Appendix B
Solution of Eq. (62) without the phenomenologically introduced a term is useful for illustrating the physics of
the result for the electric field.
Setting a equal to zero
for a neutral gas, and agreeing to explicitly solve the boundary value problem, Eq. (62) becomes (again ignoring the
4irV(V*P) term)
(v2 - 4 ^-T)-3 = % —y
c* Zt* ~
c
<B1>
*
9f
To keep the problem manageable without losing any important
features we restrict to the y-dimension.
Consider first a small volume of gas located at y' with
polarization
P(y) = 2Pr(y) cos u 0 t 6(y - y 1 ) dy
-2P±(y)sin u 0 t 6(y - y')dy
where 6 is Dirac's delta function.
(B2)
We take
E(y,y\ t) = 2A cos ojQ(t - ly ~ y ' 1 )
+ 2B sin w0(t - ly ~c Y ' I) ,
where A and B are undetermined constants.
(Bl) becomes
(B3)
When y ^ y', Eq.
188
satisfied by Eq. (B3). For y = y' take
y = y' + e
2
2
i-K
/
y = y' - e
- -A -4)Edy
ay
c^
af
4ir
2
Y •= y ' + e
= ~ (-w 0 ) /
{2Pr(y)cos u 0 t 6(y - y ' )
c
y - y' - e
- 2 P i ( y ) s i n w Q t 6(y - y ' ) }
dy
For e-K) t h e n
3y
y' + e
-^P^y')
Using Eq.
A =
4lT0)
3E
3y
3E
(2P ( y ' ) c o s to t
y'
-
e
sin a)Qt).
(B3) on t h e r i g h t hand s i d e o n e o b t a i n s
2iro)_
—
c-
Pi . (Jy ' )
(B4)
2TTO).
B =
pr(y')
Boundry surfaces have so far been neglected.
Assume now
that the dipole in Eq. (B2) is placed at coordinate y' between
Fabry-Perot mirror surfaces located at y + d/2.
The resultant
electric field is observed at - d/2 < y <. d/2.
To obtain the
required boundry condition E(y = + d/2) = 0 we use the method
of images as illustrated in Fig. 38.
Here we have the real
dipole at y' with image dipoles labelled Fn , where F is the
d
I
F2
i
F'
>
/
>
.
•
/*
1
1
/
/
/
'•
- 2 d +»'
-1
F° £F'
-d
y
/
/
/
/
/
y
-y
,
F3
L
r
0
/
i
F*
-d-
*'
"
1
-y
2d+y
3ci-y'
Figure 38. A radiating electric dipole at coordinate y' and its associated images
from two mirrors placed at y = + d/2. The resultant electric field is
observed at y.
The reflection coefficient of the mirrors is F, assumed
to be near unity.
CD
190
reflection coefficient for either mirror, and n is the number
of reflections corresponding to that particular image. One
has: real dipole,
2Pr cosu)0t - 2P. sin (jjQt, at y' ,
and image dipoles
F2lnl (2Pr cos u t - 2P. sin <0ot) at
(B5)
2nd +y', n = + 1, + 2 , ..., and
-F' 2 1 1 " 1 '
(2P,. cos a) t - 2P.
r
o
i sin coot) at
(2n - l)d-y', n = 0 , + 1 , + 2 , ...
From Eqns. (B3) and (B5), and writing down only the P. terms to
same space,
E(y, y', t) = — ° - P.(y')
c
x
{E
F^'nl
n=-»
cos Uo (t - ly - (2nd + y ) |}
-E
FI^^'COS
~. _
(B6)
co (t - ly - (2m-l)nd + y'|)}
°
c
m=-<»
+ Pr(y') terms.
Assuming the cavity to be tuned to the radiation frequency
to and using the fact that F is nearly unity, one obtains
191
4TTIO_
I
y
E(y, y', t) = — ° - (P. (y') cos m (t - l ~
c
1 *•* '
o
c
11
y
I)
+ Pr(y') sin w0(t - ly^y'l))
47TW_
__o
+
4p
(B7)
2
-E-g. (p^y*) cos w 0 t + Pr(y')sin w Q t) x
1—F
cos(kQy - IT (q/2))cos(koy' - ir (q/2)) .
Here k
= w./c.
We consider expression (B7) in two cases,
When F = 0, as in a waveguide experiment, this is
E(y, y', t) = — 3 ° - p.(y')cos wo(t - l y c y 1)
(B8)
4TTW
IV
V
I|
+ -3-°- P r ( y ' ) s i n u 0 (t - l y c y ')
.
We could have written Eq. (B2) including an arbitrary phase
term <j> so that u> t is replaced by w t + <|> in all following
results.
In the waveguide an appropriate value for <}> is
<\> = k y 1
(wave traveling from y' to y for y* ^ y) .
(B9)
In the no-boundry case then Eq. (B7) becomes
4frw
E(y, y', t) = —f-
P i cos (w0t + kQy' - kQy' + k Q y)
(BIO)
4TTCO
+ —^P- P r s i n U 0 t + k Q y ' - k Q y ' + k Q y ) ,
4irw_
= -o5-
(P
i » B ( t t o t + k Q y)
(Bll)
+ Pr sin
(w Q t + k Q y ) ) ,
192
with no y' dependence.
wave experiment P., P
We use the fact that in a traveling
are independent of y', which follows
from the derivation leading to Eq. (12) in Chapter II.
To obtain the electric field seen at the end of the
cell at y = I,
take
I
E(y = l,
4TTW
t) = / - c -°o
(P± cos (wot + kQy)
+ P r sin((oQt + kQy)) dy» ,
integrating over the source coordinate.
(B12)
The absence of this
coordinate in the integrand means that the emitted fields
add up coherently along the guide.
Now consider the F-dependent terms in Eq. (B7). These
are
47TW
o
E(y, y \ t) = — - ° -
4F2
^ - 5 - (P. (y')cos u> t
c
(B13)
l-F
+ Pr(y')sin u>0t)cos(k0y - TT (q/2) )cos (kQy' - 7r(q/2)).
Notice that E does satisfy the boundry requirements at y, y1 =
+ d/2.
Comparison of Eq. (B13) with Eq. (76) above should
make the physical reasons for the placement of the cavity
normal mode terms in Eq. (76) apparent.
Using the definition
Eq. (37) of Q one has
w
od
F2
Q = -§- — ^ T
C
(B14)
'
1 - F*
for F nearly equal to unity.
Thus
193
4F 2
1^
4Qc
=
V
(B15)
'
and Eq. (B13) becomes
E(y, y', t) =
8TTQ
| {Pi(y')cos w t
(B16)
+ Pr(y')sin wot}cos(koy - TT (q/2) )cos (koy' - Tf(q/2)).
Integrating Eq. (B16) over y' yields
E(y,t) =
8TTQ
I cos(kQy - ir(q/2)) x
d/2
/
P. (y')cos(k v' -Tr(q/2))dy'
x
-d/2
°
d/2
+ sin a) t /
P„(y')cos(k v 1 - Tr(q/2))dy'.
r
° -d/2
°
{cos to t
°
(B17)
2
The quantity u
2
VL *
°
appearing in Eq. (76) is, in one dimension
d 2
/
2
f
cos^tky - ir(q/2))dy = 1 ,
-d/2
or
u Q 2 = 2/d.
(B18)
194
BIBLIOGRAPHY
1.
C. H. Townes and A. L. Schawlow, Microwave Spectroscopy,
(McGraw-Hill, New York, 1955).
2.
R. L. Schoemaker, ''Coherent Transient Infrared Spectroscopy," Laser and Coherence Spectroscopy, Ed. by J. I.
Steinfeld (Plenum Press, N.Y., 1978).
3.
R. R. Ernst and W. A. Anderson, Rev. Sci. Inst. 37, 93
(1966) .
4.
R. H. Dicke,Phys. Rev. 93, 99 (1954).
5.
R. H. Dickeand R. H. Romer, Rev. Sci. Inst. 2_6, 915 (1955).
6.
J. C. McGurk, T. G. Schmalz, and W. H. Flygare, Adv.
Chem. Phys. XXV, 1 (1974).
7.
J. C. McGurk, R. T. Hofmann, and W. H. Flygare, J. Chem.
Phys. 60, 2922 (1974).
8.
W. H. Flygare and T. G. Schmalz, Accts. Chem. Res. 9_,
385 (1976).
9.
J. C. McGurk, H. Mader, R. T. Hofmann, T. G. Schmalz,
and W. H. Flygare, J. Chem. Phys. 61, 3759 (1974).
10.
J. Ekkers and W. H. Flygare, Rev. Sci. Inst. 47, 448
(1976) .
11.
T. G. Schmalz and W. H. Flygare, "Coherent Transient Microwave Spectroscopy," Laser and Coherence Spectroscopy,
Ed. by J. I. Sternfeld (Plenum Press, N.J., 1978), p. 125.
12.
H. Mader, J. Ekkers, W. Hoke, and W. H. Flygare, J. Chem.
Phys. 62, 4380 (1975).
13.
W. Hoke, J. Ekkers, and W. H. Flygare, J. Chem. Phys.
63, 4075 (1975).
14.
W. E. Hoke, D. R. Bauer, J. Ekkers, and W. H. Flygare,
J. Chem. Phys. 64, 5276 (1976).
15.
W. E. Hoke, D. R. Bauer, and W. H. Flygare, J. Chem.
Phys. 61, 3454 (1977).
16.
W. E. Hoke, H. L. Voss, E. J. Campbell, and W. H. Flygare, Chem. Phys. Lett. 58, 441 (1978).
195
R. E. Smalley, L. Wharton, and D. H. Levy, Accts. Chem.
Res. 10, 139 (1977).
A. C. Legon, D. J. Millen, and S. C. Rogers, Proc. R. Soc.
Lond. A 370, 213 (1980).
S. E. Novick, P. Davies, S. J. Harris, and W. Klemperer,
J. Chem. Phys. 59, 2273 (1973).
T. j . Bauer and W. H. Flygare, Res. Sci. Inst. 22, 33
(1981).
T. J. Balle, E. J. Campbell, M. R. Keenan, and W. H.
Flygare, J. Chem. Phys. 71, 2723 (1979) and 72, 922 (1980).
M. R. Keenan, E. J. Campbell, T. J. Balle, L. W. Buxton,
T. K. Minton, P. D. Soper, and W. H. Flygare, J. Chem.
Phys. 72, 3070 (1980).
E. J. Campbell, M. R. Keenan, L. W. Buxton, T. J. Balle,
P. D. Soper, A. C. Legon, and W. H. Flygare, Chem. Phys.
Lett. 70, 420 (1980).
E. J. Campbell, L. W. Buxton, T. J. Balle, and W. H.
Flygare, J. Chem. Phys. 7£, 813 (1981).
E. J. Campbell, L. W. Buxton, T. J. Balle, M. R. Keenan,
and W. H. Flygare, J. Chem. Phys. 74_, 829 (1981).
P. R. Berman, Phys. Rev. A5, 927 (1972).
H. Kogelnik and T. Li, Proc. IEEE 54_, 1312 (1966) .
M. Sargent III, M. 0. Scully, and W. E. Lamb, Jr.,
Laser Physics (Addison-Wesley, Reading, 1974).
W. E. Lamb, Jr., Phys. Rev. 134, A1429 (1964).
J.-L. LeGouet and P. R. Berman, Phys. Rev. A 20, 1105 (1979).
R. Karplus and J. Schwinger, Phys. Rev. 73_, 1020 (1948) .
E. M. Purcell, Phys. Rev. 69_, 681 (1946) .
A. C. Legon, D. J. Millen and S. C. Rogers, Chem. Phys.
Lett. 41, 137 (1976).
B. Macke, Optics Comm. 28, 131 (1979).
F. Rohart and B. Macke, J. Phys. 41, 837 (1980).
J. P. Gordon, H. J. Zeiger, and C. H. Townes, Phys. Rev.
99, 1264 (1955).
196
37.
B. L. Blaney and G. E. Ewing, Ann. Rev. Phys. Chem. 27,
553 (1976).
~
38.
G. E. Ewing, Can. J. Phys. 54_, 487 (1976).
39.
C. H. Townes and B. P. Dailey, J. Chem. Phys. 17, 782
(1949) .
40.
E. A. C. Lucken, Nuclear Quadrupole Coupling Constants
(Academic, New York, 1968).
41.
E . N. Kaufmann and R. J. Vianden, Rev. Mod. Phys. 51,
161 (1979).
42.
T. P. Das and E. L. Hahn, Nuclear Quadrupole Resonance
Spectroscopy (Academic, New York, 1958) .
43.
M. R. Keenan, L. W. Buxton, E. J. Campbell, T. J. Balle,
and W. H. Flygare, J. Chem. Phys. 73, 3523 (1980) .
44.
L. W. Buxton, E. J. Campbell, M. R. Keenan, T. J. Balle,
and W. H. Flygare, Chem. Phys. 54, 173 (1981).
45.
R. M. Sternheimer, Phys. Rev. 80, 102 (1950); 84_, 244
(1951); 86, 316 (1952); 9_5, 736 (1954).
46.
R. M. Sternheimer and H. M. Foley, Phys. Rev. 92_, 1460
(1953); 102, 731 (1956).
47.
H. M. Foley, R. M. Sternheimer, and D. Tycko, Phys. Rev.
93, 734 (1954).
48.
L. W. Buxton, E. J. Campbell, and W. H. Flygare, Chem.
Phys., (in press).
49.
E. J. Campbell, L. W. Buxton, A. C. Legon, and W. H.
Flygare, (to be published).
50.
M. R. Keenan, L. W. Buxton, E. J. Campbell, A. C. Legon,
and W.H. Flygare, J. Chem. Phys., 74, 2133 (1981).
51.
M. R. Keenan, D. B. Wozniak, and W. H. Flygare, (in
press).
52.
F. A. Baiocchi, T. A. Dixon, C. M. Joyner, and W. Klemperer,
J. Chem. Phys. (in press).
53.
J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 19 54).
54.
J. E. Gready, G. B. Bacskay and N. S. Hush, Chem. Phys.
31, 467 (1968) .
197
55.
M. R. Keenan and W. H. Flygare, Chem. Phys. Lett, (in
press).
56.
A. E. Barton, T. J. Henderson, P. R. R. Langridge-Smith,
and B. J. Howard, Chem. Phys. 45, 429 (1980) .
57.
S. E. Novick, P. Davies, S. J. Harris, and W. Klemperer,
J. Chem. Phys. 59, 2273 (1973).
58.
K. V. Chance, K. H. Bowen, J. S. Winn, and W. Klemperer,
J. Chem. Phys. 7£, 5157 (1979).
59.
W. L. Faust and L. Y. Chow Chiu, Phys. Rev. 129, 1214
(1963) .
60.
X. Husson, J-P. Grandin, and H. Kucal, J. Phys. B. 12_,
3883 (1979).
61.
P. G. Khubchandani, R. R. Sharma, and T. P. Das, Phys.
Rev. 126, 594 (1962) .
62.
R. P. McEachran, A. D. Stauffer, and S. Greita, J. Phys.
B. 12, 3119 (1979).
63.
F. D. Feiock and W. R. Johnson, Phys. Rev. 187, 39 (1969).
64.
K. D. Sen and P. T. Narasimham, Phys. Rev. B 15, 95 (1977).
65.
T. P. Das and M. Karplus, J. Chem. Phys. 42, 2885 (1965).
66.
A. Bohr, J. Koch, and E. Rasmus sen, Arkiv for Fysik 4_,
455 (1952).
67.
W. L. Faust and M. N. McDermott, Phys. Rev. 123, 198
(1961).
68.
A. E. Barton, T. J. Henderson, P. R. R. Langridge-Smith,
and B. J. Howard, Chem. Phys. 4_5 429 (1980) .
69.
K. C. Janda, L. S. Bernstein, J. M. Stted, S. E. Novick,
W. Klemperer, J. Am. Chem. Soc. 100, 8074 (1978).
70.
K. H. Bowen, K. R. Leopold, K. V. Chance, W. Klemperer,
J. Chem. Phys. 73, 137 (1980).
71.
Kenneth C. Janda, Joseph M. Steed, Stewart E. Novick,
and William Klemperer, J. Chem. Phys. 6]_, 5162 (1977) .
72.
S. E. Novick, S. J. Harris, K. C. Janda, and William
Klemperer, Can. J. Phys. 53 (1975), 2007.
E. J. Campbell, L. W. Buxton, M. R. Keenan, and W. H.
Flygare, Phys. Rev. A, (to be published).
S. J. Harris, S. E. Novick, W. Klemperer, and W. E.
Falconer, J. Chem. Phys. 61 (1974) 193.
W. Klemperer, Farad. Disc. Chem. Soc. 62 (1977) 179.
S. L. Holmgrin, M. Waldman, and W. Klemperer, J. Chem.
Phys. 67 (1977) 4414; ibid. 69 (1978) 1661.
VITA
Edward J. Campbell was born on June 28, 1954, in
Milwaukee, Wisconsin.
He graduated from the University of
Wisconsin at Madison, receiving a B.A. degree in Mathematics
and Physics in June, 1976.
He entered graduate school in
physics at the University of Illinois in June, 1976.
During
his graduate studies he has been a University Fellow, a
teaching assistant, and a research assistant.
ber of Phi Beta Kappa.
He is a mem-
He is co-author of the following
publications:
W. E. Hoke, H. L. Voss, E. J. Campbell, and W. H. Flygare,
"The Rotational Zeeman Effect in trans-Crotonaldehyde,"
Chem. Phys. Lett. 58, 441 (1978).
T. J. Balle, E. J. Campbell, M. R. Keenan, and W. H. Flygare,
"A New Method for Observing the Rotational Spectra of Weak
Molecular Complexes: KrHCl" J. Chem. Phys. 71, 2723 (1979)
and 72, 922 (1980).
E. J. Campbell, M. R. Keenan, L. W. Buxton, T. J. Balle, P.
D. Soper, A. C. Legon, and W. H. Flygare, " 83 Kr Nuclear
Quadrupole Coupling in KrHF: Evidence for Charge Transfer,"
Chem. Phys. Lett. 1Q_, 420 (1980).
M. R. Keenan, E. J. Campbell, T. J. Balle, L. W. Buxton,
T. K. Minton, P. D. Soper, and W. H. Flygare, "Rotational
Spectra and Molecular Structures of ArHBr and KrHBr," J.
Chem. Phys. 12, 3070 (1980).
M. R. Keenan, L. W. Buxton, E. J. Campbell, T. J. Balle, and
W. H. Flygare, " 1 3 1 Xe Nuclear Quadrupole Coupling and the
Rotational Spectrum of XeHCl," J. Chem. Phys. 73, 3523 (1980).
L. W. Buxton, E. J. Campbell, M. R. Keenan, T. J. Balle, and
W. H. Flygare, "The Rotational Spectrum, Nuclear Spin-Spin
Coupling, Nuclear Quadrupole Coupling, and Molecular Structure of KrHF," Chem. Phys. 54_, 173 (1981).
200
E. J. Campbell, L. W. Buxton, T. J. Balle, and W. H. Flygare,
"The Theory of Pulsed Fourier Transform Microwave Spectroscopy Carried Out in a Fabry-Perot Cavity: Static Gas,"
J. Chem. Phys. 74., 813 (1981).
E. J. Campbell, L. W. Buxton, T. J. Balle, M. R. Keenan,
and W. H. Flygare, "The Gas Dynamics of a Pulsed Supersonic
Nozzle Molecular Source as Observed with a Fabry-Perot
Cavity Microwave Spectrometer," J. Chem. Phys. 74_, 829 (1981).
M. R. Keenan, L. W. Buxton, E. J. Campbell, A. C. Legon, and
W. H. Flygare, "Molecular Structure of ArDF: An Analysis of
the Bending Mode in the Rare Gas-Hydrogen Halides," J. Chem.
Phys. 74, 2133 (1981) .
L. W. Buxton, E. J. Campbell, and W. H. Flygare, "The Vibrational Ground State Rotational Spectroscopic Constants and
Structure of the HCN Dimer," Chem. Phys. 56, 399 (1981).
E. J. Campbell, L. W. Buxton, M. R. Keenan, and W. H. Flygare,
n83 K r an{j 131xe Nuclear Quadrupole Coupling and Quadrupolar
Shielding in KrHCl and XeDCl," Phys. Rev. A. (in press).
83
L. W. Buxton, E. J. Campbell, and W. H. Flygare, " Kr
Nuclear Quadrupole Coupling in KrClF," Chem. Phys. (in press).
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